Research Article Dynamical Analysis of the Lorenz-84 ...downloads.hindawi.com/journals/jam/2014/296279.pdf · persistent abnormalities of the weather. erefore, it is very important

Research ArticleDynamical Analysis of the Lorenz-84 AtmosphericCirculation Model

Hu Wang Yongguang Yu and Guoguang Wen

Department of Mathematics Beijing Jiaotong University Beijing 100044 China

Correspondence should be addressed to Yongguang Yu ygyubjtueducn

Received 3 July 2014 Revised 15 September 2014 Accepted 16 September 2014 Published 23 November 2014

Academic Editor Qingdu Li

Copyright copy 2014 HuWang et alThis is an open access article distributed under theCreative CommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The dynamical behaviors of the Lorenz-84 atmospheric circulation model are investigated based on qualitative theory andnumerical simulationsThe stability and local bifurcation conditions of the Lorenz-84 atmospheric circulation model are obtainedIt is also shown that when the bifurcation parameter exceeds a critical value the Hopf bifurcation occurs in this model Then theconditions of the supercritical and subcritical bifurcation are derived through the normal form theory Finally the chaotic behaviorof the model is also discussed the bifurcation diagrams and Lyapunov exponents spectrum for the corresponding parameter areobtained and the parameter interval ranges of limit cycle and chaotic attractor are calculated in further Especially a computer-assisted proof of the chaoticity of the model is presented by a topological horseshoe theory

1 Introduction

Atmospheric models provided an excellent instrument forcomplex dynamical behaviors which can be observed innatural scienceThey involve processes occurring over a widespectrum of space scales and time scales from the chemistryof minor constituents in the stratosphere to hurricanesdroughts or the Quaternary glaciations and give rise to avariety of intricate behaviors in the formof abrupt transitionswave propagation weak chaos or fully developed turbu-lence [1ndash7] The generally accepted approaches to study theatmospheric and climate dynamics are based on numericalforecasting models in which all processes deemed to berelevant are included As for the low-order atmosphericmodels they involve a large number equations Althoughit is unreasonable to expect solutions to low-dimensionalproblems to generalize to a million-dimensional spaces itis unlikely too that problems identified in the simplifiedmodels will vanish in operational models [8] It is equallyimportant to note recent results which indicate the possibilitythat high-dimensional models may behave in a smooth waywith respect to changes in parameter values [9ndash11] Thuslow-order models may well have little to do with higher-dimensional operational models

On the other side the earthrsquos atmosphere is in constantcirculation due to the earthrsquos atmosphere being heated by thesun and the earthrsquos rotation A succession of the heating of theair near the earthrsquos surface rising and cool air coming downsets up a general circulation pattern air rises near the equatormoves north and south away from the equator at higheraltitudes sinks down near the poles and flows back along thesurface from both poles to the equator [12] This importanttype of flow is called Hadley circulation that was first namedafter Hadley [13] There is some evidence that the expansionof the Hadley circulation is related to climate change [14]The majority of earthrsquos driest and arid regions are locatedin the areas underneath the descending branches of theHadley circulation around 30 degrees latitude Both idealisedand more realistic climate model experiments show thatthe Hadley circulation expands with increased global meantemperature this can lead to large changes in precipitation inthe latitudes at the edge of the cells [15] Scientists fear thatthe ongoing presence of global warmingmight bring changesto the ecosystems in the deep tropics and that the deserts willbecome drier and expand [16] Based on the above discussionthe Hadley circulation is very important to the atmosphericscience Furthermore the stable and unstable atmosphericcirculations are closely linked with the dramatic changes and

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2014 Article ID 296279 15 pageshttpdxdoiorg1011552014296279

2 Journal of Applied Mathematics

persistent abnormalities of the weather Therefore it is veryimportant to research the stable and unstable atmosphericcirculation for meteorological phenomena

A very appealing low-order model of atmospheric cir-culation is introduced by Lorenz in 1984 [17] which iscalled Lorenz-84 atmospheric circulation model (Lorenz-84model) The Lorenz-84 model involves just three ordinarydifferential equations and it includes some important fea-tures of Hadley circulation So far this model was knownto have a pair of coexisting climates described originally byLorenz [18] Due to the importance of Lorenz-84model it hasreceived great attention from researchers and many impor-tant results on Lorenz-84model have also been obtained [19ndash23] In 1995 Shilrsquonikov et al discussed the bifurcation andpredictability of the Lorenz-84 model [19] Soon afterwardBroer et al studied the bifurcations and strange attractorsin the Lorenz-84 model with seasonal forcing [20] Vanet al [21] and Roebber [22] investigated the dynamicalbehaviors of a low-order coupled ocean-atmospheric modelIt is well known that the synoptic atmospheric dynamicsover the North Atlantic ocean can be dominated by the jetstream a westerly circulation and baroclinic waves whichtransport heat andmomentumnorthward Based on this factKuznetsov et al considered the intensity of the jet stream anddiscussed the fold-flip bifurcation in the Lorenz-84 model[23] For somemore detailed investigations for the Lorenz-84model the interested reader could also see [24ndash29] Howeverthe most important results are mainly based on numericalsimulations in [19 20 22 28]

This paper aims to further investigate the dynamicalbehaviors of the Lorenz-84 model by theoretical analysisSome stability conditions supercritical and subcritical Hopfbifurcations are obtained by using the qualitative analy-sis method Moreover bifurcation analysis of a nonlineardynamical system throws useful light on the behavior ofthe system in different parameter ranges Generally equi-librium points play an important role in governing theoverall system behavior It is therefore useful to consider themathematical expressions of equilibrium points as a functionof system parameters However it is so difficult to obtainthe equilibrium points regarded as the explicit mathematicalexpressions about the parameters for the Lorenz-84model Inthis paper one component of equilibrium point is regardedas a parameter and others are considered as its functionsIn this way it is not necessary to know what kind of theequilibrium point it is and the stability conditions andthe direction of the Hopf bifurcation are still obtained Inaddition there are many important results about the chaoticbehaviors of the Lorenz-84 model [17ndash20 22 28] in whichthe chaotic behaviors of the Lorenz-84 model were studiedby considering only one parameter In this paper the chaoticbehaviors of the Lorenz-84 model are studied by consideringevery parameter in the model Furthermore the topologicalhorseshoe is given in the Lorenz-84 model which providesa powerful tool in the rigorous study of chaos Finally somesimilar dynamical behaviors of the Lorenz-84 model underdifferent parameters are founded which will be very usefulfor discussing the codimension-119899 (119899 ge 2) bifurcation or othernonlinear phenomena

This paper is arranged as follows In Section 2 thestability and bifurcation of the model are discussed andthe conditions of stability and bifurcation are also givenEspecially the Hopf bifurcation is discussed and the con-ditions of the supercritical and subcritical bifurcation arederived In addition some numerical simulations are shownto verify our theoretical results and conditions The chaoticbehavior of Lorenz-84 model is researched in Section 3 Thebifurcation diagrams and Lyapunov exponent spectrum forevery parameter are discussed and the parameter intervalranges of limit cycle and chaotic attractor of every parameterare calculated Conclusions in Section 4 close the paper

2 The Stability and Bifurcation Analysisof the Lorenz-84 Model

In this section we mainly discuss the stability and localbifurcation of the Lorenz-84 model and obtain the stabilityconditions and direction of the Hopf bifurcation

21 The Lorenz-84 Model The Lorenz-84 model is a three-dimensional system [17] and is given by

119889119909

119889119905

= minus 1199102minus 1199112minus 119886119909 + 119886119865

119889119910

119889119905

= 119909119910 minus 119887119909119911 minus 119910 + 119866

119889119911

119889119905

= 119887119909119910 + 119909119911 minus 119911

(1)

where 119909 represents the strength of the globally averaged west-erly current and119910 and 119911 are the strength of the cosine and sinephases of a chain of superposedwavesTheunit of the variable119905 is equal to the damping time of the waves that is estimatedto be five days 119865 and 119866 represent the thermal forcing termsand the parameter 119887 represents the advection strength of thewaves by the westerly current Hence the equilibrium pointof model (1) satisfies the following equation

minus1199102minus 1199112minus 119886119909 + 119886119865 = 0

119909119910 minus 119887119909119911 minus 119910 + 119866 = 0

119887119909119910 + 119909119911 minus 119911 = 0

(2)

That is

119910 =

(1 minus 119909)119866

1 minus 2119909 + (1 + 1198872) 1199092

119911 =

119887119909119866

1 minus 2119909 + (1 + 1198872) 1199092

119886 (119865 minus 119909) (1 minus 2119909 + (1 + 1198872) 1199092) = 119866

2

(3)

It is well known that when the parameter 119866 = 0 thedynamical behaviors of model (1) are simple which havebeen discussed in [24] while the parameter 119866 = 0 thedynamical behaviors become complicated anddisplay chaotic

Journal of Applied Mathematics 3

attractors inmodel (1)The objective of this paper is to discussthe stability and local bifurcation and to obtain the corre-sponding stability and bifurcation conditions aboutmodel (1)with 119866 = 0 Note from (2) that it is difficult to obtain theexplicit mathematical expression of the equilibrium pointson the parameters 119886 119887 119865 and 119866 However we can alsoview the variable 119909 as a parameter Then we can get 119910 =119910(119909 119886 119887 119865 119866) 119911 = 119911(119909 119886 119887 119865 119866) and 119909 satisfies the equation119886(119865 minus 119909)(1 minus 2119909 + (1 + 119887

2)1199092) = 1198662 that are shown in (3)

22The Stability andBifurcationAnalysis with Parameters119866 =0 and 119887 = 0 When 119866 = 0 and 119887 = 0 model (1) is given as

119889119909

119889119905


119889119910

119889119905

= 119909119910 minus 119910 + 119866

119889119911

119889119905

= 119909119911 minus 119911

(4)

Then the equilibrium point of model (4) satisfies the follow-ing equation


119909119910 minus 119910 + 119866 = 0

119909119911 minus 119911 = 0

(5)

From the third equation in (5) we have (119909 minus 1)119911 = 0 If119909 = 1 then from the second equation in (5) 119866 = 0 which iscontradictory to the hypotheses 119866 = 0 in this section Henceit implies 119911 = 0 and119909 = 1Thenwe can obtain an equilibriumpoint119870(119909

0 119866(1 minus 119909

0) 0) of model (1) where 119909

0satisfies the

equation 119886(119865 minus 119909)(1 minus 119909)2 = 1198662 and 1199090= 1

Theorem 1 When 119866 = 0 and 119887 = 0 there exists at least oneequilibrium point 119870(119909

0 119866(1 minus 119909

0) 0) in model (1)

(1) if 119886 gt 0 and the equation 119886(119865 minus 119909)(1 minus 119909)2 = 1198662 hassolution 119909

0lt 1 the equilibrium point 119870 is stable


0gt 1 the equilibrium point 119870 is unstable

(3) if 119886 lt 0 and the equation 119886(119865 minus 119909)(1 minus 119909)2 = 1198662 hassolution 1 minus 3radicminus21198662119886 lt 119909

0lt 1 + 119886 the equilibrium

point 119870 is stable

Proof The Jacobian matrix of model (1) at the point119870(1199090 119866(1 minus 119909

0) 0) is given as

119869 = (

minus119886 minus211991000

11991001199090minus 1 0

0 0 1199090minus 1

) (6)

and its characteristic equation is given as

119891 (120582) = (120582 + 1 minus 1199090) (1205822+ (119886 + 1 minus 119909

0) 120582

+ (119886 + 1) 1199090+ 21199102

0)

(7)

Then eigenvalues 1205821 1205822 and 120582

3are obtained as follows

1205821= 1199090minus 1

1205822=

minus (119886 + 1 minus 1199090) + radic(119886 minus 1 + 119909

0)2

minus (81198662 (1 minus 119909

0)2

)

2

1205823=

minus (119886 + 1 minus 1199090) minus radic(119886 minus 1 + 119909

0)2

minus (81198662 (1 minus 119909

0)2

)

2

(8)

If 119886 gt 0 and the equation 119886(119865 minus 119909)(1 minus 119909)2 = 1198662 has solution1199090lt 1 then there is 120582

1lt 0 Since 81198662(1 minus 119909

0)2gt 0 and

(119886 + 1 minus 1199090)2minus (119886 minus 1 + 119909

0)2= 4(1 minus 119909

0)119886 gt 0 it follows

that Re(1205822) lt 0 and Re(120582

3) lt 0 Therefore the equilibrium

point119870 is stable If the solution 1199090gt 1 it is easy to verify that

the equilibrium point119870 is unstable If 119886 lt 0 and the equation119886(119865 minus 119909)(1 minus 119909)

2= 1198662 has solution 1 minus 3radicminus21198662119886 lt 119909

0lt

1 + 119886 it follows that Re(1205821) lt 0 Re(120582

2) lt 0 and Re(120582

3) lt

0 Therefore the equilibrium point 119870 is stable The proof iscompleted

Theorem 2 If 119866 = 0 and 119887 = 0 then there exists at least oneequilibrium point 119870(119909

0 119866(1 minus 119909

0) 0) in model (1)

(1) when 119886 gt 0 and 21198662 = 1198864 if 1199090= 119886 + 1 is a solution

of the equation 119886(119865 minus 119909)(1 minus 119909)2 = 1198662 the equilibriumpoint119870(119909

0 119866(1minus119909

0) 0) is unstable andmodel (1) has

a fold bifurcation

(2) when 119886 lt 0 and 21198662 gt 1198864 if 1199090= 119886 + 1 is a solution


0 119866(1minus119909


a Hopf bifurcation

Proof It is easy to obtain the result from (8) and is thereforeomitted here

23 Direction and Stability of theHopf Bifurcation In this sec-tion the supercritical and subcritical bifurcations of model(1) are considered According to Theorem 2 model (1) has aHopf bifurcation when 119909

0= 119886 + 1 21198662 gt 1198864 and 119886 lt 0 Let

the eigenvectors corresponding to the eigenvalues 1205821and 120582

3

be 1205721+ 1198941205722and 120572

3 respectively where 120572

1 1205722 and 120572

3are all

real vectors Under the following linear transformation

1199091= 119909 minus 119909

0

1199101= 119910 minus 119910

0

1199111= 119911 minus 119911

0

(9)


model (1) can be changed into

1= minus 119910

2

1minus 211991001199101minus 1199102

0minus 1199112

1+ 1198861199090minus 1198861199091+ 119886119865

1199101= 11990911199101+ 11990911199100+ 11990901199101+ 11990901199100minus 1199101minus 1199100+ 119866

1= 11990911199111+ 11990901199111minus 1199111

(10)

For (6) (7) and (8) if 1199090= 119886 + 1 is a solution of the equation

119886(119865 minus 119909)(1 minus 119909)2= 1198662 21198662gt 1198864 and 119886 lt 0 we have

1205721015840(0) = Re (1205821015840 (119886)) = 1

2

gt 0

1205961015840(0) = Im (1205821015840 (119886)) =

radic41198662minus 1198864

4 (21198662minus 1198864)

= 0

120596 (0) = Im (120582 (119886)) =radic21198662minus 1198864

|119886|

gt 0

1205823lt 0

(11)

Therefore model (1) at the equilibrium point 119870 has a Hopfbifurcation By calculations we have

1205721= (

1

0

0

) 1205722= (

119886

119870

119866

119886119870

0

) 1205723= (

0

0

1

)

119879 = (1205722 1205721 1205723) = (

119886

119896

1 0

119866

119886119896

0 0

0 0 1

)

(12)

where 119896 = radic21198662 minus 1198864119886For model (10) taking the following transformation

(

119909111991011199111

) = 119879 (

119909211991021199112

) we have

1199092= 1198601199092+ 119861 (13)

where

119860 = (

0 minus119896 0

119896 0 0

0 0 119886

) 1199092= (

1199092

1199102

1199112

)

119861 =(

119886

119896

1199092

2+ 11990921199102+

119886119896

119866

11990901199100minus 1199100+ 119866

minus

1198662+ 1198864

119886211989621199092

2minus

119886

119896

11990921199102minus 1199112

2minus (

1198862

119866

+

2119866

119886119896

)11990901199100minus 1198861199090+

1198862

119866

1199090+ 119886119865 minus 119886

2

119886

119896

11990921199112+ 11991021199112

)

(14)

From (13)

2= minus 119896119910

2+ 119875 (119909

2 1199102 1199112)

1199102= 1198961199092+ 119876 (119909

2 1199102 1199112)

2= 1198861199112+ 119877 (119909

2 1199102 1199112)

(15)

where

119875 (1199092 1199102 1199112) =

119886

119896

1199092

2+ 11990921199102+

119886119896

119866

11990901199100minus 1199100+ 119866

119876 (1199092 1199102 1199112)

= minus

1198662+ 1198864

119886211989621199092

2minus

119886

119896

11990921199102minus 1199112

2

minus (

1198862

119866

+

2119866

119886119896

)11990901199100minus 1198861199090+

1198862

119866

1199090+ 119886119865 minus 119886

2

119877 (1199092 1199102 1199112) =

119886

119896

11990921199112+ 11991021199112

(16)

For model (15) according to the normal form theory in [3031] we have

11989211=

1

4

(

1205971198752

1205971199092

2

+

1205971198752

1205971199102

2

+ 119894 (

1205971198762

1205971199092

2

+

1205971198762

1205971199102

2

))

=

1

2

(

119886

119896

minus

1198662+ 1198864

11988621198962119894)

11989202=

1

4

(

1205971198752

1205971199092

2

minus

1205971198752

1205971199102

2

minus 2

1205971198762

12059711990921205971199102

+ 119894 (

1205971198762

1205971199092

2

minus

1205971198762

1205971199102

2

+ 2

1205971198752

12059711990921205971199102

))

=

1

2

(1 minus

1198662+ 1198864

11988621198962) 119894

11989220=

1

4

(

1205971198752

1205971199092

2

minus

1205971198752

1205971199102

2

+ 2

1205971198762

12059711990921205971199102

+ 119894 (

1205971198762

1205971199092

2

minus

1205971198762

1205971199102

2

minus 2

1205971198752

12059711990921205971199102

))


=

1

2

(minus1 minus

1198662+ 1198864

11988621198962) 119894

11986621=

1

8

(

1205971198753

1205971199093

2

+

1205971198753

12059711990921205971199102

2

+

1205971198763

1205971199092

21205971199102

+

1205971198763

1205971199103

2

+ 119894 (

1205971198763

1205971199093

2

+

1205971198763

12059711990921205971199102

2

minus

1205971198753

1205971199092

21205971199102

minus

1205971198753

1205971199103

2

)) = 0

(17)

Since the dimension 119899 = 3 gt 2 we obtain the followingequations

ℎ11=

1

4

(

1205971198772

1205971199092

2

+

1205971198772

1205971199102

2

) = 0

ℎ20=

1

4

(

1205971198772

1205971199092

2

minus

1205971198772

1205971199102

2

minus 2

1205971198772

12059711990921205971199102

) = 0

(18)

Next we solve the following equations

120582112060111= minusℎ11

(1205821minus 2119896119894) 120601

20= minusℎ20

(19)

Then

12060111= 0

12060120= 0

(20)

Let

119866110=

1

2

(

1205971198752

12059711990921205971199112

+

1205971198762

12059711991021205971199112

+ 119894 (

1205971198752

12059711990921205971199112

minus

1205971198762

12059711991021205971199112

)) = 0

119866101=

1

2

(

1205971198752

12059711990921205971199112

minus

1205971198762

12059711991021205971199112

+ 119894 (

1205971198752

12059711990921205971199112

+

1205971198762

12059711991021205971199112

)) = 0

(21)

Taking119898 = (1198662 + 1198864)11988621198962 it then follows from (17) and (21)that

11989221= 11986621+ (2119866

11012060111+ 11986610112060120) = 0 (22)

Also let

1198721(0) =

119894

2119896

(1198922011989211minus 2100381610038161003816100381611989211

1003816100381610038161003816

2

minus

1

3

100381610038161003816100381611989202

1003816100381610038161003816

2

) +

1

2

11989221

=

(119898 + 1) 119886

81198962minus

119894

8119896

(

21198862

1198962+

10

3

1198982+

1

3

119898 +

1

3

)

(23)

Then we have

1205832(119886 119887) = minus

Re (1198721(0))

1205721015840(0)

= minus

(119898 + 1) 119886

41198962gt 0 (119886 lt 0)

1205732(119886 119887) = 2Re (119872

1(0)) =

(119898 + 1) 119886

41198962lt 0 (119886 lt 0)

1205912(119886 119887) = minus

Im (1198721(0)) + 120583

2(119886 119887) 120596

1015840(0)

119887

= minus(

1

81198962(

21198862

1198962+

10

3

1198982+

1

3

119898 +

1

3

)

minus

(119898 + 1) 119886

161198962

radic41198662minus 1198864

4 (21198662minus 1198864)

) lt 0

(24)

Theorem 3 When 21198662 gt 1198864 119886 lt 0 there is a Hopfbifurcation in model (1) at 119909

0= 119886 + 1 and

(1) 1205832(119886 119887) gt 0 that is to say the direction of bifurcation

is supercritical(2) 1205732(119886 119887) lt 0 that is to say the solutions of bifurcating

periodic solutions are orbitally stable(3) 1205912(119886 119887) lt 0 that is to say the periods of bifurcating

periodic solutions increase

Proof It is easy to obtain the result from the above derivativeprocess in Section 23 and is therefore omitted here

24 The Stability and Bifurcation Analysis with Parameters119866 = 0 and 119887 = 0 In this section we consider the stabilityand bifurcation analysis of model (1) when 119866 = 0 and 119887 = 0Without loss of generality we first discuss the stability andbifurcation analysis of model (1) at two special equilibriumpoints when 119909 = 0 and 119909 = 1 It is easy to verify that when119909 = 0 and 119909 = 1 the equilibrium points of model (1) are1198701(0 119866 0) and119870

2(1 0 119866119887) respectively

In the following let us consider the stability and bifurca-tion analysis of model (1) at the equilibrium point119870

1

Theorem 4 For model (1) if 119886 gt 0 the equilibrium point 1198701

is stable and(1) when 119886 ge 1 + 2radic2119866 gt 0 or 0 lt 119886 le 1 minus 2radic2119866 the

equilibrium point 1198701is stable node point

(2) when 0 lt 1 minus 2radic2119866 lt 119886 lt 1 + 2radic2119866 the equilibriumpoint 119870

1is stable focus point

Proof For model (1) the Jacobian matrix at the equilibriumpoint119870

1is

119869 = (

minus119886 minus2119866 0

119866 minus1 0

119887119866 0 minus1

) (25)

and its characteristic equation is

119891 (120582) = (120582 + 1) (1205822+ (119886 + 1) 120582 + 119886 + 2119866

2) (26)


We can get 120582123

1205821= minus1

12058223=

minus (119886 + 1) plusmn radic(119886 + 1)2minus 4 (119886 + 2119866

2)

2

(27)

Since 119886 gt 0 it then follows from (27) that Re(120582123) lt 0

Therefore the equilibrium point 1198701is stable In the same

time note that if Δ = (119886 + 1)2 minus 4(119886 + 21198662) gt 0 120582123lt 0 are

real numbers and it then follows that the equilibrium point1198701is stable node point if Δ = (119886 + 1)2 minus 4(119886 + 21198662) lt

0 12058223

has conjugate imaginary roots it then follows thatthe equilibrium point 119870

1is stable focus point The proof is

completed

Theorem 5 For model (1) if 119886 lt 0 and

(1) when 119886 = minus1 119865 lt minus12 and 1198662 gt 12 then model (1)at the equilibrium point 119870

1has a Hopf bifurcation

(2) when 119886 = minus1 119865 = minus2 and 1198662 = 2 then model (1) atthe equilibrium point 119870

1has a Fold bifurcation

(3) when 119886 = minus1 119865 = minus12 then model (1) at theequilibrium point 119870

1has a flip bifurcation

Proof According to (27) we can easily get the conclusiontherefore we omit it here

Next let us consider the equilibrium point 1198702(1 0 119866119887)

Theorem 6 If 119886 gt 0 and 119865 lt 32 then the equilibrium point1198702of model (1) is stable

Proof The Jacobian matrix of model (1) at the equilibriumpoint119870

2is

119869 = (

minus119886 minus2119866 minus2

119866

119887

119866 0 119887

minus

119866

119887

119887 0

) (28)


119891 (120582) = 1205823+ 1198861205822+ (1198872+

21198662

1198872)120582 + 119886119887

2minus 21198662 (29)

Let 119860 = 119886 119861 = 1198872 + (211986621198872) and 119862 = 1198861198872 minus 21198662 By theRouth-Hurwitz criterion when119860 gt 0119862 gt 0 and119860119861minus119862 gt 0the eigenvalues of (29) have negative real partsTherefore wecan get 119860119861 minus 119862 = 21198662(1 + (1198861198872)) gt 0 119862 gt 0 and 119865 lt 32Therefore the equilibrium point119870

2ofmodel (1) is stableThe

proof is completed

In the above discussions we consider the stability andbifurcation analysis of model (1) at two special equilibriumpoints 119870

1and 119870

2 respectively In the following we will

discuss stability and bifurcation analysis of model (1) at thegeneral equilibrium point119870(119909

0 1199100 1199110)

Theorem 7 Suppose 119870(1199090 1199100 1199110) is an equilibrium point in

model (1) if 1199090is a solution of the equation 119886(119865 minus 119909)(1 minus 2119909 +

(1198872+ 1)1199092) = 1198662 and satisfies the inequations

1199090lt 1 +

119886

2

1199092

0+

4 + 2119865 + 21198651198872

3 (1 + 1198872)

1199090+

1 + 2119865

3 (1 + 1198872)

gt 0

1199093

0minus

5119886 + 21198872minus 1198861198872+ 6

2 (1 + 1198872)

1199092

0+

9119886 + 6 + 119886119865 + 31198862minus 1198861198872119865

2 (1 + 1198872)

1199090

minus

(1 + 119886) (2 + 2119886 + 119886119865)

2 (1 + 1198872)

lt 0

(30)

then the equilibrium point 119870 is stable

Proof From (2)

1199100= 119892 (119909

0 119887) (1 minus 119909

0) 119866

1199110= 119892 (119909

0 119887) 1199090119887119866

119886 (119865 minus 119909) 119891 (1199090 119887) = 119866

2

(31)

where119891(1199090 119887) = 1minus2119909

0+(1198872+1)1199092

0and119892(119909

0 119887) = 1119891(119909

0 119887)

The Jacobian matrix of model (1) at the point 119870(1199090 1199100 1199110) is

given as

119869 (0) =(

minus119886 minus2119892 (1199090 119887) (1 minus 119909

0) 119866 minus2119892 (119909

0 119887) 119887119909

0119866

119892 (1199090 119887) (1 minus 119909

0minus 11988721199092

0)119866 119909

0minus 1 minus119887119909

0

119887119892 (1199090 119887) 119866 119887119909

01199090minus 1

) (32)


Taking 119866119892(1199090 119887) = 119879 we can get

119891 (120582) = 1205823+ (2 minus 2119909

0+ 119886) 120582

2

+ (2119886 (1 minus 1199090) + 119891 (119909

0 119887) (1 + 2119879

2)) 120582

+ 119891 (1199090 119887) (119886 + 2119879

2(1 minus 119909

0minus 11988721199090))

(33)

Let 119860 = 2 minus 21199090+ 119886 119861 = 2119886(1 minus 119909

0) + 119891(119909

0 119887)(1 + 2119879

2) and

119862 = 119891(1199090 119887)(119886 + 2119879

2(1 minus 119909

0minus 11988721199090))

Note that the 119891(1199090 119887) gt 0 119892(119909

0 119887) gt 0 1198792 gt 0 and

1199090lt 1 +

119886

2

1199092

0+

4 + 2119865 + 21198651198872

3 (1 + 1198872)

1199090+

1 + 2119865

3 (1 + 1198872)

gt 0

1199093

0minus

5119886 + 21198872minus 1198861198872+ 6

2 (1 + 1198872)

1199092

0

+

9119886 + 6 + 119886119865 + 31198862minus 1198861198872119865

2 (1 + 1198872)

1199090

minus

(1 + 119886) (2 + 2119886 + 119886119865)

2 (1 + 1198872)

lt 0

(34)

and we have 119860 gt 0 119862 gt 0 and 119860119861 minus 119862 gt 0 According to theRouth-Hurwitz criterion the real parts of the roots of (33) areall negative Therefore the equilibrium point 119870 is stable Theproof is completed

Theorem8 Suppose that119870(1199090 1199100 1199110) is an equilibrium point

in model (1) If 1199090is a solution of the equation 119886(119865 minus 119909)(1 minus

2119909 + (1198872+ 1)1199092) = 119866

2 and satisfies the following inequationsand equation

1199090lt 1 +

119886

2

1199092

0+

4 + 2119865 + 21198651198872

3 (1 + 1198872)

1199090+

1 + 2119865

3 (1 + 1198872)

gt 0

1199093

0minus

5119886 + 21198872minus 1198861198872+ 6

2 (1 + 1198872)

1199092

0

+

9119886 + 6 + 119886119865 + 31198862minus 1198861198872119865

2 (1 + 1198872)

1199090

minus

(1 + 119886) (2 + 2119886 + 119886119865)

2 (1 + 1198872)

= 0

12057311199094

0minus 12057321199093

0+ 12057331199092

0+ 12057341199090+ 1205735= 0

(35)

then model (1) has a Hopf bifurcation where

1205731=

41198862(1 minus 119887

4)

1198662

1205732=

41198862

1198662(2 + 2119887

2+ 119886 + 119886119887

2+ 2119865 minus 2119865119887

4)

1205733=

411988621198652(1 minus 119887

4)

1198662

+

81198651198862(2 + 2119887

2+ 119886 + 119886119887

2)

1198662

+ 3 (1198872+ 1)

1205734= 4119886119887

2minus 14119886 minus 8 minus 4119887

2minus

41198862119865 (2 + 2119887

2+ 119886 + 119886119887

2)

1198662

1205735= 12119886 + 2119886119865 + 2119886119887

2119865 + 4119886

2+ 3

(36)

Proof From (33) we can obtain 119860 = 2 minus 21199090+ 119886 119861 = 2119886(1 minus

1199090)+119891(119909

0 119887)(1+2119879

2) and119862 = 119891(119909

0 119887)(119886+2119879

2(1minus1199090minus11988721199090))

Since

1199090lt 1 +

119886

2

1199092

0+

4 + 2119865 + 21198651198872

3 (1 + 1198872)

1199090+

1 + 2119865

3 (1 + 1198872)

gt 0

1199093

0minus

5119886 + 21198872minus 1198861198872+ 6

2 (1 + 1198872)

1199092

0

+

9119886 + 6 + 119886119865 + 31198862minus 1198861198872119865

2 (1 + 1198872)

1199090

minus

(1 + 119886) (2 + 2119886 + 119886119865)

2 (1 + 1198872)

= 0

(37)

it follows that 119860(1199090) gt 0 119861(119909

0) = 119862(119909

0)119860(119909

0) gt 0 and

119862(1199090) gt 0 Similarly since 120573

11199094

0minus12057321199093

0+12057331199092

0+12057341199090+1205735= 0

it follows that 1198621015840(1199090) = 1198601015840(1199090)119861(1199090) minus 119860(119909

0)1198611015840(1199090) where

1205731=

41198862(1 minus 119887

4)

1198662

1205732=

41198862

1198662(2 + 2119887

2+ 119886 + 119886119887

2+ 2119865 minus 2119865119887

4)

1205733=

411988621198652(1 minus 119887

4)

1198662

+

81198651198862(2 + 2119887

2+ 119886 + 119886119887

2)

1198662

+ 3 (1198872+ 1)

1205734= 4119886119887


2

minus

41198862119865 (2 + 2119887

2+ 119886 + 119886119887

2)

1198662

1205735= 12119886 + 2119886119865 + 2119886119887

2119865 + 4119886

2+ 3

(38)


Hence we have proven that 119860(1199090) gt 0 119861(119909

0) =

119862(1199090)119860(119909

0) gt 0 119862(119909

0) gt 0 and 1198621015840(119909

0) = 119860

1015840(1199090)119861(1199090) minus

119860(1199090)1198611015840(1199090) Therefore according to the Hopf bifurcation

theorem [30] model (1) has a Hopf bifurcation The proof iscompleted

25 Simulation In this section some numerical examplesand simulations are presented to illustrate the effectivenessof our theoretical results Here we mainly discuss and verifythe conditions of Theorems 1 2 and 3

Firstly we verify the effectiveness of Theorem 1 with 119886 gt0 Taking 119886 = 025 119865 = 8 then the curves of 119886(119865 minus119909)(1 minus 119909)

2= 1198662 and straight line of 119909 = 1 are shown

in Figure 1(a) in which 119909-axis represents the parameter 119866and 119910-axis represents 119909 values of the equilibrium point119870(119909 119866(1 minus 119909) 0) Note that when 119886 = 025 119865 = 8 and119866 = 3 there are three equilibrium points 119860 = (minus1 15 0)119861 = (7 minus05 0) and 119862 = (4 minus1 0) which are shown inFigure 1(a) From Figure 1(b) the converging trajectories forpoints 119861 and119862 are Heteroclinic orbits and they will convergeto the equilibrium point 119860 Hence the equilibrium posits 119861119862 are unstable Take an arbitrary point119863 (see Figure 1) underthe conditions 119886 = 025 119865 = 8 and 119866 = 3 It is shown fromFigure 1(b) that the trajectory starting from119863 converges to119860Therefore we have verified that the point 119860 is stable

In addition we consider 119886 lt 0 in Theorem 1 Choose119886 = minus025 119865 = minus6 and then the curves of 119886(119865 minus119909)(1 minus 119909)

2= 1198662 and straight line of 119909 = 1 are shown in

Figure 2(a) When 119886 = minus025 119865 = minus6 and 119866 = 3 wecan obtain three equilibrium points 119860(minus2 1 0) 119861(3 minus15 0)and 119862(minus5 05 0) which are shown in Figure 2(a) FromFigure 2(b) the converging trajectories for points119861 and119862 areheteroclinic orbits and they will converge to the equilibriumpoint 119860 Hence the equilibrium posits 119861 119862 are unstable andthe equilibrium posits 119860 is stable

Next we verify the effectiveness ofTheorems 2 and 3 Takethe parameters 119866 = 2 119865 = minus4 and 119886 = minus1 which satisfy theHopf bifurcation conditions 1198863(119865minus119886minus1) = 1198662 and 21198662 gt 1198864Figure 3(a) shows the phase plane of model (1) with 119866 = 19119865 = minus4 and 119886 = minus1 It can be seen from Figure 3(a) thatthere is a stable limit cycle Figure 3(b) shows the phase planeof model (1) with 119866 = 21 119865 = minus4 and 119886 = minus1 It canbe seen from Figure 3(b) that there is a stable equilibriumpointTherefore we have verified that model (1) exists a Hopfbifurcation In addition we can calculate 120583

2(119886 119887) = 17

according toTheorem 5 andHopf bifurcation is supercriticalAlso we can calculate 120573

2(119886 119887) = minus94radic7 lt 0 according to

Theorem 5 and the periodic solution is stable

3 Chaotic Behavior of the Lorenz-84 Model

In this section the chaotic behavior of model (1) with theparameters 119866 119887 119886 and 119865 are discussed and the complexdynamic behaviors are analyzed by Lyapunov exponentsspectrum bifurcation diagram Poincare section and powerspectrum Especially the topological horseshoe is given torigorous approaches to study chaos in model (1) For the sake

of simplicity the initial condition in model (1) is chosen as[1 1 1]

31 Dynamical Behaviors of Model (1) by Varying Parameters119866 and 119887 This model has been found to be chaotic over awide range of parameters and has many interesting complexdynamical behaviors by varying parameters 119866 and 119887 respec-tively It is well known that Lyapunov exponents measurethe exponential rates of divergence or convergence of nearbytrajectories in phase spaceThedynamical behaviors ofmodel(1) by varying parameters 119866 and 119887 are listed in Table 1

Dynamical behaviors of model (1) by varying parameter119866 are first discussed Suppose that the parameters 119886 = 025119865 = 8 and 119887 = 4 are fixed Let the parameter 119866 varyin interval [0 14] The Lyapunov exponents spectrum isshown in Figure 4 It is shown from Figure 4 that when 119866 isin[0831 1188] and 119866 isin [0831 1188] the max Lyapunovexponents are greater than zero that is to say model (1) haschaotic state Figure 5(a) shows the bifurcation diagram ofmodel (1) about the parameter 119866 Figure 5(b) shows the partof Figure 5(a) with119866 gt 0 It is shown fromFigure 5 that when119866 isin [0 14] model (1) can evolve into chaotic stateMoreoverthere are some periodic windows in the chaotic regions Itis shown from Figure 5(a) that the dynamical behaviors ofmodel (1) have the symmetry for the parameter 119866 that is tosay when the parameters 119866 gt 0 and 119866 lt 0 model (1) has thesame dynamical behaviors

To illustrate the chaotic behavior the parameter 119866 ischosen as 119866 = 1 The corresponding phase portrait ofchaotic attractor is shown in Figure 6(a) and its spectrum iscontinuous as shown in Figure 6(b) FromFigure 6model (1)has chaotic state Next the topological horseshoe will be usedto study chaos in model (1)The topological horseshoe is wellrecognized as one of the most rigorous approaches to studychaos with computer It has been successfully applied inmanychaotic systems and hyperchaotic systems [32ndash40] As a basicand striking result in chaotic dynamics topological horse-shoe provides a powerful tool in the rigorous study of chaosand dynamical systems obtaining the topological entropyverifying the existence of chaos showing the structure ofchaotic attractors revealing the mechanism inside chaoticphenomena and so on Some definitions and propertiesabout the topological horseshoe [33ndash36] are given at first

Let 119863 be a compact connected region of 119877119899 and let 119863119894

119894 = 1 2 119898 be disjoint compact connected subsets of 119863homeomorphic to the unit square Let 119891 119863 rarr 119877119899 bea piecewise continuous map which is a diffeomorphism oneach compact set119863

119894

Definition 9 (see [33 36]) For each 119863119894 1 le 119894 le 119898 let 1198631

119894

and1198632119894be two fixed disjoint compact subsets of119863

119894contained

in the boundary 120597119863119894 A connected subset 119897 of 119863

119894is said to

connect1198631119894and1198632

119894 if 119897 cap 1198631

119894= and 119897 cap 1198632

119894=

Definition 10 (see [33 36]) Let 119897 sub 119863119894be a connection of

1198631

119894and 1198632

119894 We say that 119891(119897) is acrossing 119863

119895 if 119897 contains a

connected subset 1198971015840 such that119891(1198971015840) sub 119863119895is a connection of1198631

119895

and1198632119895 In this case we denote it by 119891 997891rarr 119863

119895 Furthermore if


Table 1 The dynamic behaviors of model (1) about the parameters 119866 119887 119886 and 119865

The parameter 119866 Dynamic behaviors The parameter 119887 Dynamic behaviors(119886 = 025 119887 = 4 and 119865 = 8) (119886 = 025 119865 = 8 and 119866 = 1)0 lt 119866 lt 0277 Limit cycle 0 lt 119887 lt 107 Limit point0277 le 119866 lt 0797 Limit cycle period 2 107 le 119887 lt 173 Limit cycle0797 le 119866 lt 0801 Limit cycle period 4 173 le 119887 lt 253 Chaos0801 le 119866 lt 0819 Limit cycle period 2 253 le 119887 lt 393 Limit cycle0819 le 119866 lt 0831 Limit cycle period 4 393 le 119887 lt 416 Chaos0831 le 119866 lt 1188 Chaos 416 le 119887 lt 553 Limit cycle1188 le 119866 lt 1212 Limit cycle period 4 553 le 119887 lt 59 Chaos1212 le 119866 lt 1352 Chaos 59 le 119887 lt 853 Limit cycle1352 le 119866 lt 1364 Limit cycle period 2 853 le 119887 lt 860 Chaos1364 le 119866 lt 185 Limit point 860 le 119887 lt 10 Limit cycleThe parameter 119886 Dynamic behaviors The parameter 119865 Dynamic behaviors(119887 = 4 119865 = 8 and 119866 = 1) (119886 = 025 119887 = 4 and 119866 = 1)0 lt 119886 lt 0134 Limit point 0 lt 119865 lt 4313 Limit point0134 le 119886 lt 0152 Chaos 4313 le 119865 lt 4518 Limit cycle0152 le 119886 lt 0207 Limit cycle 4518 le 119865 lt 5159 Chaos0207 le 119886 lt 0240 Chaos 5159 le 119865 lt 6948 Limit cycle0240 le 119886 lt 0243 Limit cycle 6948 le 119865 lt 8763 Chaos0243 le 119886 lt 0280 Chaos 8763 le 119865 lt 15 Limit cycle0280 le 119886 lt 03 Limit cycle

0 2 4 6

0

2

4

6

8

G

x

C

B

A

minus6 minus4 minus2minus4

minus2

x = 1

(a)

02

46

8

01

23

45

68

ZY

DC

A

BX024

minus6

minus4

minus2

minus2minus1

(b)

Figure 1 (a) The curve of 119886(119865 minus 119909)(1 minus 119909)2 = 1198662 (b) Phase plane of system (1) with the parameter 119866 = 3

119891 997891rarr 119863119895holds true for every connection 119897 of1198631

119894and1198632

119894 then

119891(119863119894) is said to be acrossing119863

119895and is denoted by119891(119863

119894) sub 119863

119895

with respect to two pairs (1198631119894 1198632

119894) and (1198631

119895 1198632

119895)

Theorem 11 (see [34ndash36]) If the relation119891(119863119894) sub 119863

119895holds for

every pair with 119894 119895 taken from 1 le 119894 119895 le 119898 then there exists acompact invariant set119870 sub 119863 such that 119891 | 119870 is semiconjugateto the full 119898-shift dynamics 120590 | sum

119898and topological entropy

119890119899119905(119891) ge log119898

Here the 119898-shift is also called the Bernoulli 119898-shifThe symbolic series space sum

119898is compact totally discon-

nected and perfect More definitions and properties aboutthe topological horseshoe can be found in [32ndash40] To find

horseshoes in model (1) we use the method proposed in[34] with an efficient and powerful toolbox in MATLABcalled ldquoA toolbox for finding horseshoes in 2D maprdquo (Li Q DHSTOOL-A MATLAB toolbox for finding horseshoes in 2Dmaps 2007ndash2011 httpwwwmathworkscnmatlabcentralfileexchange14075) When 119866 = 1 according to the Matlabtoolbox we can get Poincare map of model (1) at the 119911 = 0(Figure 7(a)) We numerically find two quadrilateral 119863

1and

1198632in the 119909minus119910 (Poincare map) planeThe four vertices of119863

1

are(0727230483 1444887781) (0666821561 1360099751)

(0843401487 1245386534) (0913104089 1330174564)

(39)


0

2

4

x A

C

B

minus6

minus4

minus2

0 2 4 6G


minus8

x = 1

(a)

0 1 2 3005

115

ZY

X

0 2

005

115

Z

Y

A

B

C

0

2

4

minus6

minus4

minus2

minus2

minus2minus15minus1

minus05

minus15minus1

minus1

minus05

minus3

(b)

Figure 2 (a) The curve of 119886(119865 minus 119909)(1 minus 119909)2 = 1198662 (b) Phase plane of of system (1) with the parameter 119866 = 3

005

1

01

23

45

0123

zy

x

minus1minus05

minus4

minus3

minus2

minus1

(a)

005

1

01

23

45

0123

zy

x

minus1minus05

minus4

minus3

minus2

minus1

(b)

Figure 3 (a) Phase plane of system (1) with parameter 119866 = 19 (b) Phase plane of system (1) with parameters 119866 = 21

0 02 04 06 08 1 12 14

0

02

04

G

Lyap

unov

expo

nent

s

minus12

minus1

minus08

minus06

minus04

minus02

Figure 4 Lyapunov exponents about 119866 for the parameter 119865 = 8

and the four vertices of1198632are

(1029275093 1245386534) (0964219331 1165586035)

(1085037175 1040897756) (1164033457 1130673317)

(40)

Based on the theoretical result [32ndash40] andTheorem 11

Theorem 12 There exists a compact invariant set 119870 sub 1198631cup

1198632 such that1198675 | 119870 is semiconjugate to 2-shift dynamics and

the topological entropy of119867 is 119890119899119905(119867) ge (15) log 2

Proof According to Theorem 11 we need to show that therelations

1198675(1198631) 997891997888rarr 119863

1 119867

5(1198631) 997891997888rarr 119863

2

1198675(1198632) 997891997888rarr 119863

1 119867

5(1198632) 997891997888rarr 119863

2

(41)

hold true For the above four relations it is easy to see fromFigure 7(b) Furthermore it also follows from Theorem 11that the topological entropy of 119867 is ent(119867) ge (15) log 2Clearly ent(119867) gt 0 which means that model (1) is chaoticindeed when 119866 = 1 The proof is completed

In [17] when the parameters 119886 and 119887 are respectivelychosen as 119886 = 025 and 119887 = 4 Lorenz pointed out that119865 and 119866 should be allowed to vary periodically during oneyear In particular 119865 should be larger in winter than insummer In the numerical study of [17] the author pointedthat (119865 119866) = (6 1) and (119865 119866) = (8 1) represent the summerconditions and the winter conditions respectively With thewinter conditions (119865 119866) = (8 1) model (1) has chaotic


28

26

24

22

2

18

16

14

12

1minus1 minus05 0 05 1

G

x

(a)

26

24

22

2

18

16

12

14

06 07 08 09 1 11 12 13 14 15

G

x

(b)

Figure 5 The bifurcation diagram about 119866

minus20

24

minus20

24

0

1

2

3

Z

Y

X

minus1

minus4 minus4

(a)

0 200 400 600 800 1000

0102030405060

Pow

er sp

ectr

um

minus40

minus30

minus20

minus10

10 lowast log10(S)

(b)

Figure 6 (a) Phase portraits of the chaos attractor (b) Spectrum of 119909

0 05 1 15 2 25

0

05

1

15

2

25

x

y

minus25

minus2

minus15

minus1

minus05

minus05

D1

D2

(a)

0 02 04 06 08 1 12 14 16

04

06

08

1

12

14

16

18

x

y

07 08 09 1 111

11

12

13

14

15

Zoom in

D1

D2

D2

D1

D11

D11

D12

D12

D21

D21

D22

H5(D1)

H5(D1)

H5(D2)

H5(D2)

H5(D12)

H5(D11)

H5(D22)

H5(D21)

(b)

Figure 7 (a) Poincare mapping on the crossing 119911 = 0 (b) The topological horseshoe at 119866 = 1


0 02 04 06 08 1 12 14 16 18 2

0

05

G

Lyap

unov

expo

nent

s

minus3

minus25

minus2

minus15

minus05

minus1

(a) Lyapunov exponents about 119866 for the parameter 119865 = 6

0 02 04 06 08 1 12 14 16 18 2

0

05

G

Lyap

unov

expo

nent

s

minus1

minus25

minus2

minus15

minus05

(b) Lyapunov exponents about 119866 for the parameter 119865 = 10

Figure 8 Lyapunov exponents about 119866

0 1 2 3 4 5 6 7 8 9 10

0

02

b

Lyap

unov

expo

nent

s

minus08

minus06

minus04

minus02


0

05

1

15

2

25

3

35

4

x

b

minus10 minus8 minus6 minus4 minus2 0 2 4 6 8 10

(a)

0

05

1

15

2

25

3

35

4

45

x

b

2 4 6 81 3 5 7

(b)



01 015 02 025 03

0

05

a

Lyap

unov

expo

nent

s

minus05

minus1

minus15

minus2

(a)

4 45 5 55 6 65 7 75 8 85 9

0

02

04

F

Lyap

unov

expo

nent

s

minus02

minus04

minus06

minus08

minus1

minus12

(b)

Figure 11 (a) Lyapunov exponents about 119886 (b) Lyapunov exponents about 119865

0

05

1

15

2

25

3

35

4

45

a

x

minus0501 012 014 016 018 02 022 024 026 028 03

(a)

0

05

1

15

2

25

3

35

F

x

4 45 5 55 6 65 7 75 8 85 9 95

(b)

Figure 12 (a) The bifurcation diagram about 119886 (b) The bifurcation diagram about 119865

behavior In this paper 119886 and 119887 are respectively chosen as025 and 4 and 119865 is chosen as 6 7 8 9 and 10 respectivelyAccording to the numerical study model (1) appears to be ofmore complex dynamics behavior when119865 = 8 thanwhen119865 =6 7 9 10 by varying parameter 119866 Moreover model (1) hasbeen found to be chaotic over more wider range of parameter119866 when 119865 = 8 than when 119865 = 6 7 9 10 respectively When119865 = 8 the Lyapunov exponent spectra have been alreadyshown in Figure 4 In the following we consider the cases of119865 = 6 and 119865 = 10 respectively The cases of 119865 = 7 and 119865 = 9are similar to the cases of 119865 = 6 and 119865 = 10 therefore theyare omitted here When the parameter 119865 = 6 and 119865 = 10the Lyapunov exponents about 119866 are shown in Figures 8(a)and 8(b) respectively It is shown that from Figure 8(a) thatmodel (1) has no chaotic behavior when 119865 = 6 And from

Figure 8(b) when119865 = 10 model (1) has the chaotic behaviorswith 1473 le 119866 lt 1538

Dynamical behaviors of model (1) by varying parameters119887 are also considered When parameters 119886 = 025 119865 = 8 and119866 = 1 are fixed let the parameter 119887 vary in interval [0 10]Similarly the Lyapunov exponents spectrum is shown inFigure 9 It is shown from Figure 9 that when 119887 isin [173 253]119887 isin [393 416] 119887 isin [553 590] and 119887 isin [853 860]the max Lyapunov exponents are greater than zero thatis to say model (1) has chaotic state The correspondingbifurcation diagram is displayed in Figure 10 It is shown fromFigure 10(a) that the dynamical behaviors of model (1) havethe symmetry for the parameter 119887 that is to say when theparameter 119887 gt 0 and 119887 lt 0 model (1) has the same dynamicalbehaviors Figure 10(b) shows the part of Figure 10(a) with


119887 gt 0 This is similar to the parameter 119866 and the dynamicalbehaviors of model (1) about parameter 119887 are more keenlyaware of the change than the parameter 119866

32 Dynamical Behaviors of Model (1) by Varying Parameters119886 and 119865 This section mainly focuses on the dynamicalbehaviors of model (1) by varying parameters 119886 and 119865respectivelyThe dynamical behaviors ofmodel (1) by varyingparameters 119886 and 119865 are also listed in Table 1 Dynamicalbehaviors of model (1) by varying parameters 119886 are firstconsidered When parameters 119887 = 4 119865 = 8 and 119866 = 1are fixed let the parameter 119886 varies in interval [01 03] theLyapunov exponents spectrum is shown in Figure 11(a) Itis shown from Figure 11(a) that when 119886 isin [01345 01526]119886 isin [0207 02406] and 119886 isin [02435 02806] and themax Lyapunov exponents are greater than zero model (1)has chaotic state The corresponding bifurcation diagram isdisplayed in Figure 12(a)

Similarly when parameters 119886 = 025 119887 = 4 and119866 = 1 arefixed let the parameter 119865 vary in interval [4 9] the Lyapunovexponents spectrum is shown in Figure 11(b) When 119865 isin[4518 5159] and 119865 isin [6948 8763] the max Lyapunovexponents are greater than zero and model (1) has chaoticstate Figure 12(b) shows the bifurcation diagram of model(1) about the parameter 119865 From the Lyapunov exponentsspectrum and the bifurcation diagram shown in Figures 11and 12 the dynamical behaviors of model (1) about 119886 and 119865are very similar

4 Conclusion

In this paper the stability and local bifurcation of the Lorenz-84 model have been investigated The stability and localbifurcation of model (1) with 119866 = 0 have been discussedThe conditions of the supercritical Hopf bifurcation have alsobeen derived In addition the chaotic behaviors of Lorenz-84 model have been researched The bifurcation diagramsand Lyapunov exponents spectrums for every parameterhave been discussed and the parameter interval range oflimit cycle and chaotic attractor of every parameter havealso been calculated Especially a computer-assisted proof ofthe chaoticity of the Lorenz-84 model has been presentedby a topological horseshoe theory Future work will focuson multistability and high codimension bifurcations of theLorenz-84 model

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This paper is supported by the National Nature ScienceFoundation of China (no 11371049) and Science Foundationof BJTU (2014YJS132)

References

[1] R Benzi A Sutera and A Vulpiani ldquoThe mechanism ofstochastic resonancerdquo Journal of Physics A Mathematical andGeneral vol 14 no 11 pp L453ndashL457 1981

[2] R Benzi G Parisi A Sutera and A Vulpiani ldquoStochasticresonance in climate changerdquo Tellus vol 34 pp 10ndash16 1982

[3] C Nicolis ldquoSolar variability and stochastic effects on climaterdquoSolar Physics vol 74 no 2 pp 473ndash478 1981

[4] C Nicolis ldquoStochastic aspects of climatic transitions-responseto a periodic forcingrdquo Tellus vol 34 pp 1ndash9 1982

[5] C Nicolis and G Nicolis Irreversible Phenomena and Dynami-cal Systems Analysis in Geosciences D Reidel Publishing 1987

[6] G Nicolis and C Nicolis Foundations of Complex SystemsWorld Scientific Singapore 2007

[7] C Nicolis and G Nicolis ldquoStability complexity and the max-imum dissipation conjecturerdquo Quarterly Journal of the RoyalMeteorological Society vol 136 no 650 pp 1161ndash1169 2010

[8] L A Smith ldquoWhat might we learn from climate forecastsrdquoProceedings of the National Academy of Sciences of the UnitedStates of America vol 99 no 1 pp 2487ndash2492 2002

[9] D J Albers and J C Sprott ldquoStructural stability and hyper-bolicity violation in high-dimensional dynamical systemsrdquoNonlinearity vol 19 no 8 pp 1801ndash1847 2006

[10] D J Albers J C Sprott and J P Crutchfield ldquoPersistent chaos inhigh dimensionsrdquo Physical Review E Statistical Nonlinear andSoft Matter Physics vol 74 no 5 Article ID 057201 2006

[11] V Lucarini A Speranza andRVitolo ldquoParametric smoothnessand self-scaling of the statistical properties of aminimal climatemodel what beyond the mean field theoriesrdquo Physica DNonlinear Phenomena vol 234 no 2 pp 105ndash123 2007

[12] G Hadley ldquoOn the cause of the general trade windsrdquo Philosoph-ical Transactions of the Royal Society vol 34 pp 58ndash62 1735

[13] A Persson ldquoHadleyrsquos principle understanding and misunderstanding the trade windsrdquo History of Meteorology vol 3 pp 17ndash42 2006

[14] F D Henry and S B RaymondTheHadley Circulation PresentPast and Future vol 21 of Advances in Global Change ResearchSpringer Amsterdam The Netherlands 2004

[15] D M W Frierson J Lu and G Chen ldquoWidth of the Hadleycell in simple and comprehensive general circulation modelsrdquoGeophysical Research Letters vol 34 no 18 Article ID L188042007

[16] D J Seidel Q Fu W J Randel and T J Reichler ldquoWidening ofthe tropical belt in a changing climaterdquo Nature Geoscience vol1 no 1 pp 21ndash24 2008

[17] E N Lorenz ldquoIrregularity a fundamental property of theatmosphererdquo Tellus A vol 36 no 2 pp 98ndash110 1984

[18] E N Lorenz ldquoCan chaos and intransitivity lead to interannualvariabilityrdquo Tellus Series A vol 42 no 3 pp 378ndash389 1990

[19] A Shilrsquonikov G Nicolis and C Nicolis ldquoBifurcation andpredictability analysis of a low-order atmospheric circulationmodelrdquo International Journal of Bifurcation and Chaos inApplied Sciences and Engineering vol 5 no 6 pp 1701ndash17111995

[20] H Broer C Simo and R Vitolo ldquoBifurcations and strangeattractors in the Lorenz-84 climate model with seasonal forc-ingrdquo Nonlinearity vol 15 no 4 pp 1205ndash1267 2002

[21] V L Van T Opsteegh and F Verhulst ldquoThe dynamics of a loworder coupled ocean-atmosphere modelrdquo Journal of NonlinearScience httparxivorgabschao-dyn9812024


[22] P J Roebber ldquoClimate variability in a low-order coupledatmosphere-ocean modelrdquo Tellus A vol 47 no 4 pp 473ndash4941995

[23] Y A Kuznetsov H G E Meijer and L van Veen ldquoThe fold-flipbifurcationrdquo International Journal of Bifurcation and Chaos inApplied Sciences and Engineering vol 14 no 7 pp 2253ndash22822004

[24] YG Yu ldquoDynamical analysis of a low-ordermodel representingHadley circulationrdquo Beijingjiaotong University vol 3 pp 50ndash542006 (Chinese)

[25] Y Song Y Yu and H Wang ldquoThe stability and chaos analysisof the Lorenz-84 atmosphere model with seasonal forcingrdquo inProceedings of the 4th InternationalWorkshop on Chaos-FractalsTheories and Applications (IWCFTA rsquo11) pp 37ndash41 October2011

[26] L van Veen ldquoBaroclinic flow and the Lorenz-84 modelrdquo Inter-national Journal of Bifurcation and Chaos in Applied Sciencesand Engineering vol 13 no 8 pp 2117ndash2139 2003

[27] L Niklas ldquoBifurcations and strange attractors in a climaterelated systemrdquoDifferential Equations and Control Processes no1 pp 1ndash53 2005

[28] J G Freire C Bonatto C C DaCamara and J A Gallas ldquoMul-tistability phase diagrams and intransitivity in the Lorenz-84low-order atmospheric circulation modelrdquo Chaos vol 18 no 3Article ID 033121 8 pages 2008

[29] V Pelino andA Pasini ldquoDissipation in Lie-poisson systems andthe Lorenz-84 modelrdquo Physics Letters A General Atomic andSolid State Physics vol 291 no 6 pp 389ndash396 2001

[30] B D Hassard N D Kazarinoff and Y Wan Theory andApplications of Hopf Bifurcation vol 41 Cambridge UniversityPress Cambridge UK 1981

[31] Y Yu and S Zhang ldquoHopf bifurcation in the Lu systemrdquo ChaosSolitons and Fractals vol 17 no 5 pp 901ndash906 2003

[32] X-S Yang ldquoTopological horseshoes in continuous mapsrdquoChaos Solitons amp Fractals vol 33 no 1 pp 225ndash233 2007

[33] X-S Yang ldquoTopological horseshoes and computer assistedverification of chaotic dynamicsrdquo International Journal of Bifur-cation andChaos in Applied Sciences and Engineering vol 19 no4 pp 1127ndash1145 2009

[34] Q Li and X S Yang ldquoA simple method for finding topologicalhorseshoesrdquo International Journal of Bifurcation and Chaos inApplied Sciences and Engineering vol 20 no 2 pp 467ndash4782010

[35] Q Li X-S Yang and S Chen ldquoHyperchaos in a spacecraftpower systemrdquo International Journal of Bifurcation and Chaosvol 21 no 6 pp 1719ndash1726 2011

[36] Q Li S Tang and X-S Yang ldquoNew bifurcations in the simplestpassive walking modelrdquo Chaos vol 23 no 4 Article ID 0431102013

[37] S Huan Q Li and X-S Yang ldquoHorseshoes in a chaoticsystem with only one stable equilibriumrdquo International Journalof Bifurcation and Chaos in Applied Sciences and Engineeringvol 23 no 1 Article ID 1350002 2013

[38] Q D Li and S Tang ldquoAlgorithm for finding horseshoes inthree-dimensional hyperchaotic maps and its applicationrdquoActaPhysica Sinica vol 62 no 2 Article ID 020510 2013

[39] Q D Li S Tang and X-S Yang ldquoHyperchaotic set incontinuous chaos-hyperchaos transitionrdquo Communications inNonlinear Science and Numerical Simulation vol 19 no 10 pp3718ndash3734 2014

[40] Q Li andX S Yang ldquoTwo kinds of horseshoes in a hyperchaoticneural networkrdquo International Journal of Bifurcation and Chaosvol 22 no 8 Article ID 1250200 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

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Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


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OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


persistent abnormalities of the weather Therefore it is veryimportant to research the stable and unstable atmosphericcirculation for meteorological phenomena

A very appealing low-order model of atmospheric cir-culation is introduced by Lorenz in 1984 [17] which iscalled Lorenz-84 atmospheric circulation model (Lorenz-84model) The Lorenz-84 model involves just three ordinarydifferential equations and it includes some important fea-tures of Hadley circulation So far this model was knownto have a pair of coexisting climates described originally byLorenz [18] Due to the importance of Lorenz-84model it hasreceived great attention from researchers and many impor-tant results on Lorenz-84model have also been obtained [19ndash23] In 1995 Shilrsquonikov et al discussed the bifurcation andpredictability of the Lorenz-84 model [19] Soon afterwardBroer et al studied the bifurcations and strange attractorsin the Lorenz-84 model with seasonal forcing [20] Vanet al [21] and Roebber [22] investigated the dynamicalbehaviors of a low-order coupled ocean-atmospheric modelIt is well known that the synoptic atmospheric dynamicsover the North Atlantic ocean can be dominated by the jetstream a westerly circulation and baroclinic waves whichtransport heat andmomentumnorthward Based on this factKuznetsov et al considered the intensity of the jet stream anddiscussed the fold-flip bifurcation in the Lorenz-84 model[23] For somemore detailed investigations for the Lorenz-84model the interested reader could also see [24ndash29] Howeverthe most important results are mainly based on numericalsimulations in [19 20 22 28]

This paper aims to further investigate the dynamicalbehaviors of the Lorenz-84 model by theoretical analysisSome stability conditions supercritical and subcritical Hopfbifurcations are obtained by using the qualitative analy-sis method Moreover bifurcation analysis of a nonlineardynamical system throws useful light on the behavior ofthe system in different parameter ranges Generally equi-librium points play an important role in governing theoverall system behavior It is therefore useful to consider themathematical expressions of equilibrium points as a functionof system parameters However it is so difficult to obtainthe equilibrium points regarded as the explicit mathematicalexpressions about the parameters for the Lorenz-84model Inthis paper one component of equilibrium point is regardedas a parameter and others are considered as its functionsIn this way it is not necessary to know what kind of theequilibrium point it is and the stability conditions andthe direction of the Hopf bifurcation are still obtained Inaddition there are many important results about the chaoticbehaviors of the Lorenz-84 model [17ndash20 22 28] in whichthe chaotic behaviors of the Lorenz-84 model were studiedby considering only one parameter In this paper the chaoticbehaviors of the Lorenz-84 model are studied by consideringevery parameter in the model Furthermore the topologicalhorseshoe is given in the Lorenz-84 model which providesa powerful tool in the rigorous study of chaos Finally somesimilar dynamical behaviors of the Lorenz-84 model underdifferent parameters are founded which will be very usefulfor discussing the codimension-119899 (119899 ge 2) bifurcation or othernonlinear phenomena

This paper is arranged as follows In Section 2 thestability and bifurcation of the model are discussed andthe conditions of stability and bifurcation are also givenEspecially the Hopf bifurcation is discussed and the con-ditions of the supercritical and subcritical bifurcation arederived In addition some numerical simulations are shownto verify our theoretical results and conditions The chaoticbehavior of Lorenz-84 model is researched in Section 3 Thebifurcation diagrams and Lyapunov exponent spectrum forevery parameter are discussed and the parameter intervalranges of limit cycle and chaotic attractor of every parameterare calculated Conclusions in Section 4 close the paper

2 The Stability and Bifurcation Analysisof the Lorenz-84 Model

In this section we mainly discuss the stability and localbifurcation of the Lorenz-84 model and obtain the stabilityconditions and direction of the Hopf bifurcation

21 The Lorenz-84 Model The Lorenz-84 model is a three-dimensional system [17] and is given by

119889119909

119889119905


119889119910

119889119905

= 119909119910 minus 119887119909119911 minus 119910 + 119866

119889119911

119889119905

= 119887119909119910 + 119909119911 minus 119911

(1)

where 119909 represents the strength of the globally averaged west-erly current and119910 and 119911 are the strength of the cosine and sinephases of a chain of superposedwavesTheunit of the variable119905 is equal to the damping time of the waves that is estimatedto be five days 119865 and 119866 represent the thermal forcing termsand the parameter 119887 represents the advection strength of thewaves by the westerly current Hence the equilibrium pointof model (1) satisfies the following equation


119909119910 minus 119887119909119911 minus 119910 + 119866 = 0

119887119909119910 + 119909119911 minus 119911 = 0

(2)

That is

119910 =

(1 minus 119909)119866

1 minus 2119909 + (1 + 1198872) 1199092

119911 =

119887119909119866

1 minus 2119909 + (1 + 1198872) 1199092

119886 (119865 minus 119909) (1 minus 2119909 + (1 + 1198872) 1199092) = 119866

2

(3)

It is well known that when the parameter 119866 = 0 thedynamical behaviors of model (1) are simple which havebeen discussed in [24] while the parameter 119866 = 0 thedynamical behaviors become complicated anddisplay chaotic



2)1199092) = 1198662 that are shown in (3)


119889119909

119889119905


119889119910

119889119905

= 119909119910 minus 119910 + 119866

119889119911

119889119905

= 119909119911 minus 119911

(4)



119909119910 minus 119910 + 119866 = 0

119909119911 minus 119911 = 0

(5)


0 119866(1 minus 119909

0) 0) of model (1) where 119909

0satisfies the



0 119866(1 minus 119909

0) 0) in model (1)









0) 0) is given as

119869 = (

minus119886 minus211991000

11991001199090minus 1 0

0 0 1199090minus 1

) (6)


119891 (120582) = (120582 + 1 minus 1199090) (1205822+ (119886 + 1 minus 119909

0) 120582

+ (119886 + 1) 1199090+ 21199102

0)

(7)



1205821= 1199090minus 1

1205822=


0)2

minus (81198662 (1 minus 119909

0)2

)

2

1205823=


0)2

minus (81198662 (1 minus 119909

0)2

)

2

(8)


1lt 0 Since 81198662(1 minus 119909

0)2gt 0 and


0)2= 4(1 minus 119909


that Re(1205822) lt 0 and Re(120582





0lt


2) lt 0 and Re(120582

3) lt



0 119866(1 minus 119909

0) 0) in model (1)



0 119866(1minus119909


a fold bifurcation



0 119866(1minus119909


a Hopf bifurcation



0= 119886 + 1 21198662 gt 1198864 and 119886 lt 0 Let


3

be 1205721+ 1198941205722and 120572


1 1205722 and 120572

3are all


1199091= 119909 minus 119909

0

1199101= 119910 minus 119910

0

1199111= 119911 minus 119911

0

(9)



1= minus 119910

2

1minus 211991001199101minus 1199102

0minus 1199112

1+ 1198861199090minus 1198861199091+ 119886119865

1199101= 11990911199101+ 11990911199100+ 11990901199101+ 11990901199100minus 1199101minus 1199100+ 119866

1= 11990911199111+ 11990901199111minus 1199111

(10)



1205721015840(0) = Re (1205821015840 (119886)) = 1

2

gt 0

1205961015840(0) = Im (1205821015840 (119886)) =

radic41198662minus 1198864

4 (21198662minus 1198864)

= 0

120596 (0) = Im (120582 (119886)) =radic21198662minus 1198864

|119886|

gt 0

1205823lt 0

(11)


1205721= (

1

0

0

) 1205722= (

119886

119870

119866

119886119870

0

) 1205723= (

0

0

1

)

119879 = (1205722 1205721 1205723) = (

119886

119896

1 0

119866

119886119896

0 0

0 0 1

)

(12)


(

119909111991011199111

) = 119879 (

119909211991021199112

) we have

1199092= 1198601199092+ 119861 (13)

where

119860 = (

0 minus119896 0

119896 0 0

0 0 119886

) 1199092= (

1199092

1199102

1199112

)

119861 =(

119886

119896

1199092

2+ 11990921199102+

119886119896

119866

11990901199100minus 1199100+ 119866

minus

1198662+ 1198864

119886211989621199092

2minus

119886

119896

11990921199102minus 1199112

2minus (

1198862

119866

+

2119866

119886119896

)11990901199100minus 1198861199090+

1198862

119866

1199090+ 119886119865 minus 119886

2

119886

119896

11990921199112+ 11991021199112

)

(14)

From (13)

2= minus 119896119910

2+ 119875 (119909

2 1199102 1199112)

1199102= 1198961199092+ 119876 (119909

2 1199102 1199112)

2= 1198861199112+ 119877 (119909

2 1199102 1199112)

(15)

where

119875 (1199092 1199102 1199112) =

119886

119896

1199092

2+ 11990921199102+

119886119896

119866

11990901199100minus 1199100+ 119866

119876 (1199092 1199102 1199112)

= minus

1198662+ 1198864

119886211989621199092

2minus

119886

119896

11990921199102minus 1199112

2

minus (

1198862

119866

+

2119866

119886119896

)11990901199100minus 1198861199090+

1198862

119866

1199090+ 119886119865 minus 119886

2

119877 (1199092 1199102 1199112) =

119886

119896

11990921199112+ 11991021199112

(16)


11989211=

1

4

(

1205971198752

1205971199092

2

+

1205971198752

1205971199102

2

+ 119894 (

1205971198762

1205971199092

2

+

1205971198762

1205971199102

2

))

=

1

2

(

119886

119896

minus

1198662+ 1198864

11988621198962119894)

11989202=

1

4

(

1205971198752

1205971199092

2

minus

1205971198752

1205971199102

2

minus 2

1205971198762

12059711990921205971199102

+ 119894 (

1205971198762

1205971199092

2

minus

1205971198762

1205971199102

2

+ 2

1205971198752

12059711990921205971199102

))

=

1

2

(1 minus

1198662+ 1198864

11988621198962) 119894

11989220=

1

4

(

1205971198752

1205971199092

2

minus

1205971198752

1205971199102

2

+ 2

1205971198762

12059711990921205971199102

+ 119894 (

1205971198762

1205971199092

2

minus

1205971198762

1205971199102

2

minus 2

1205971198752

12059711990921205971199102

))


=

1

2

(minus1 minus

1198662+ 1198864

11988621198962) 119894

11986621=

1

8

(

1205971198753

1205971199093

2

+

1205971198753

12059711990921205971199102

2

+

1205971198763

1205971199092

21205971199102

+

1205971198763

1205971199103

2

+ 119894 (

1205971198763

1205971199093

2

+

1205971198763

12059711990921205971199102

2

minus

1205971198753

1205971199092

21205971199102

minus

1205971198753

1205971199103

2

)) = 0

(17)


ℎ11=

1

4

(

1205971198772

1205971199092

2

+

1205971198772

1205971199102

2

) = 0

ℎ20=

1

4

(

1205971198772

1205971199092

2

minus

1205971198772

1205971199102

2

minus 2

1205971198772

12059711990921205971199102

) = 0

(18)


120582112060111= minusℎ11

(1205821minus 2119896119894) 120601

20= minusℎ20

(19)

Then

12060111= 0

12060120= 0

(20)

Let

119866110=

1

2

(

1205971198752

12059711990921205971199112

+

1205971198762

12059711991021205971199112

+ 119894 (

1205971198752

12059711990921205971199112

minus

1205971198762

12059711991021205971199112

)) = 0

119866101=

1

2

(

1205971198752

12059711990921205971199112

minus

1205971198762

12059711991021205971199112

+ 119894 (

1205971198752

12059711990921205971199112

+

1205971198762

12059711991021205971199112

)) = 0

(21)


11989221= 11986621+ (2119866

11012060111+ 11986610112060120) = 0 (22)

Also let

1198721(0) =

119894

2119896

(1198922011989211minus 2100381610038161003816100381611989211

1003816100381610038161003816

2

minus

1

3

100381610038161003816100381611989202

1003816100381610038161003816

2

) +

1

2

11989221

=

(119898 + 1) 119886

81198962minus

119894

8119896

(

21198862

1198962+

10

3

1198982+

1

3

119898 +

1

3

)

(23)

Then we have

1205832(119886 119887) = minus

Re (1198721(0))

1205721015840(0)

= minus

(119898 + 1) 119886

41198962gt 0 (119886 lt 0)

1205732(119886 119887) = 2Re (119872

1(0)) =

(119898 + 1) 119886

41198962lt 0 (119886 lt 0)

1205912(119886 119887) = minus

Im (1198721(0)) + 120583

2(119886 119887) 120596

1015840(0)

119887

= minus(

1

81198962(

21198862

1198962+

10

3

1198982+

1

3

119898 +

1

3

)

minus

(119898 + 1) 119886

161198962

radic41198662minus 1198864

4 (21198662minus 1198864)

) lt 0

(24)


0= 119886 + 1 and







2(1 0 119866119887) respectively


1







1is

119869 = (

minus119886 minus2119866 0

119866 minus1 0

119887119866 0 minus1

) (25)


119891 (120582) = (120582 + 1) (1205822+ (119886 + 1) 120582 + 119886 + 2119866

2) (26)



1205821= minus1

12058223=


2)

2

(27)





0 12058223



completed












2is

119869 = (


119866

119887

119866 0 119887

minus

119866

119887

119887 0

) (28)


119891 (120582) = 1205823+ 1198861205822+ (1198872+

21198662

1198872)120582 + 119886119887

2minus 21198662 (29)



proof is completed


1and 119870



0 1199100 1199110)




1199090lt 1 +

119886

2

1199092

0+

4 + 2119865 + 21198651198872

3 (1 + 1198872)

1199090+

1 + 2119865

3 (1 + 1198872)

gt 0

1199093

0minus

5119886 + 21198872minus 1198861198872+ 6

2 (1 + 1198872)

1199092

0+

9119886 + 6 + 119886119865 + 31198862minus 1198861198872119865

2 (1 + 1198872)

1199090

minus

(1 + 119886) (2 + 2119886 + 119886119865)

2 (1 + 1198872)

lt 0

(30)


Proof From (2)

1199100= 119892 (119909

0 119887) (1 minus 119909

0) 119866

1199110= 119892 (119909

0 119887) 1199090119887119866

119886 (119865 minus 119909) 119891 (1199090 119887) = 119866

2

(31)

where119891(1199090 119887) = 1minus2119909

0+(1198872+1)1199092

0and119892(119909

0 119887) = 1119891(119909

0 119887)


given as

119869 (0) =(


0) 119866 minus2119892 (119909

0 119887) 119887119909

0119866

119892 (1199090 119887) (1 minus 119909

0minus 11988721199092

0)119866 119909

0minus 1 minus119887119909

0

119887119892 (1199090 119887) 119866 119887119909

01199090minus 1

) (32)


Taking 119866119892(1199090 119887) = 119879 we can get

119891 (120582) = 1205823+ (2 minus 2119909

0+ 119886) 120582

2

+ (2119886 (1 minus 1199090) + 119891 (119909

0 119887) (1 + 2119879

2)) 120582

+ 119891 (1199090 119887) (119886 + 2119879

2(1 minus 119909

0minus 11988721199090))

(33)

Let 119860 = 2 minus 21199090+ 119886 119861 = 2119886(1 minus 119909

0) + 119891(119909

0 119887)(1 + 2119879

2) and

119862 = 119891(1199090 119887)(119886 + 2119879

2(1 minus 119909

0minus 11988721199090))

Note that the 119891(1199090 119887) gt 0 119892(119909

0 119887) gt 0 1198792 gt 0 and

1199090lt 1 +

119886

2

1199092

0+

4 + 2119865 + 21198651198872

3 (1 + 1198872)

1199090+

1 + 2119865

3 (1 + 1198872)

gt 0

1199093

0minus

5119886 + 21198872minus 1198861198872+ 6

2 (1 + 1198872)

1199092

0

+

9119886 + 6 + 119886119865 + 31198862minus 1198861198872119865

2 (1 + 1198872)

1199090

minus

(1 + 119886) (2 + 2119886 + 119886119865)

2 (1 + 1198872)

lt 0

(34)




2119909 + (1198872+ 1)1199092) = 119866


1199090lt 1 +

119886

2

1199092

0+

4 + 2119865 + 21198651198872

3 (1 + 1198872)

1199090+

1 + 2119865

3 (1 + 1198872)

gt 0

1199093

0minus

5119886 + 21198872minus 1198861198872+ 6

2 (1 + 1198872)

1199092

0

+

9119886 + 6 + 119886119865 + 31198862minus 1198861198872119865

2 (1 + 1198872)

1199090

minus

(1 + 119886) (2 + 2119886 + 119886119865)

2 (1 + 1198872)

= 0

12057311199094

0minus 12057321199093

0+ 12057331199092

0+ 12057341199090+ 1205735= 0

(35)


1205731=

41198862(1 minus 119887

4)

1198662

1205732=

41198862

1198662(2 + 2119887

2+ 119886 + 119886119887

2+ 2119865 minus 2119865119887

4)

1205733=

411988621198652(1 minus 119887

4)

1198662

+

81198651198862(2 + 2119887

2+ 119886 + 119886119887

2)

1198662

+ 3 (1198872+ 1)

1205734= 4119886119887


2minus

41198862119865 (2 + 2119887

2+ 119886 + 119886119887

2)

1198662

1205735= 12119886 + 2119886119865 + 2119886119887

2119865 + 4119886

2+ 3

(36)


1199090)+119891(119909

0 119887)(1+2119879

2) and119862 = 119891(119909

0 119887)(119886+2119879

2(1minus1199090minus11988721199090))

Since

1199090lt 1 +

119886

2

1199092

0+

4 + 2119865 + 21198651198872

3 (1 + 1198872)

1199090+

1 + 2119865

3 (1 + 1198872)

gt 0

1199093

0minus

5119886 + 21198872minus 1198861198872+ 6

2 (1 + 1198872)

1199092

0

+

9119886 + 6 + 119886119865 + 31198862minus 1198861198872119865

2 (1 + 1198872)

1199090

minus

(1 + 119886) (2 + 2119886 + 119886119865)

2 (1 + 1198872)

= 0

(37)


0) = 119862(119909

0)119860(119909

0) gt 0 and


11199094

0minus12057321199093

0+12057331199092

0+12057341199090+1205735= 0


0)1198611015840(1199090) where

1205731=

41198862(1 minus 119887

4)

1198662

1205732=

41198862

1198662(2 + 2119887

2+ 119886 + 119886119887

2+ 2119865 minus 2119865119887

4)

1205733=

411988621198652(1 minus 119887

4)

1198662

+

81198651198862(2 + 2119887

2+ 119886 + 119886119887

2)

1198662

+ 3 (1198872+ 1)

1205734= 4119886119887


2

minus

41198862119865 (2 + 2119887

2+ 119886 + 119886119887

2)

1198662

1205735= 12119886 + 2119886119865 + 2119886119887

2119865 + 4119886

2+ 3

(38)



0) =

119862(1199090)119860(119909

0) gt 0 119862(119909

0) gt 0 and 1198621015840(119909

0) = 119860

1015840(1199090)119861(1199090) minus











2(119886 119887) = 17












119894


119894


119894contained


119894is said to

connect1198631119894and1198632

119894 if 119897 cap 1198631

119894= and 119897 cap 1198632

119894=


1198631

119894and 1198632




119895






0 2 4 6

0

2

4

6

8

G

x

C

B

A


minus2

x = 1

(a)

02

46

8

01

23

45

68

ZY

DC

A

BX024

minus6

minus4

minus2

minus2minus1

(b)



119894and1198632

119894 then


119895and is denoted by119891(119863

119894) sub 119863

119895


119894) and (1198631

119895 1198632

119895)


119895holds for



119890119899119905(119891) ge log119898





1and


1

are(0727230483 1444887781) (0666821561 1360099751)

(0843401487 1245386534) (0913104089 1330174564)

(39)


0

2

4

x A

C

B

minus6

minus4

minus2

0 2 4 6G


minus8

x = 1

(a)

0 1 2 3005

115

ZY

X

0 2

005

115

Z

Y

A

B

C

0

2

4

minus6

minus4

minus2

minus2

minus2minus15minus1

minus05

minus15minus1

minus1

minus05

minus3

(b)


005

1

01

23

45

0123

zy

x

minus1minus05

minus4

minus3

minus2

minus1

(a)

005

1

01

23

45

0123

zy

x

minus1minus05

minus4

minus3

minus2

minus1

(b)


0 02 04 06 08 1 12 14

0

02

04

G

Lyap

unov

expo

nent

s

minus12

minus1

minus08

minus06

minus04

minus02



(1029275093 1245386534) (0964219331 1165586035)

(1085037175 1040897756) (1164033457 1130673317)

(40)






1198675(1198631) 997891997888rarr 119863

1 119867

5(1198631) 997891997888rarr 119863

2

1198675(1198632) 997891997888rarr 119863

1 119867

5(1198632) 997891997888rarr 119863

2

(41)




28

26

24

22

2

18

16

14

12


G

x

(a)

26

24

22

2

18

16

12

14

06 07 08 09 1 11 12 13 14 15

G

x

(b)


minus20

24

minus20

24

0

1

2

3

Z

Y

X

minus1

minus4 minus4

(a)

0 200 400 600 800 1000

0102030405060

Pow

er sp

ectr

um

minus40

minus30

minus20

minus10

10 lowast log10(S)

(b)


0 05 1 15 2 25

0

05

1

15

2

25

x

y

minus25

minus2

minus15

minus1

minus05

minus05

D1

D2

(a)

0 02 04 06 08 1 12 14 16

04

06

08

1

12

14

16

18

x

y

07 08 09 1 111

11

12

13

14

15

Zoom in

D1

D2

D2

D1

D11

D11

D12

D12

D21

D21

D22

H5(D1)

H5(D1)

H5(D2)

H5(D2)

H5(D12)

H5(D11)

H5(D22)

H5(D21)

(b)



0 02 04 06 08 1 12 14 16 18 2

0

05

G

Lyap

unov

expo

nent

s

minus3

minus25

minus2

minus15

minus05

minus1


0 02 04 06 08 1 12 14 16 18 2

0

05

G

Lyap

unov

expo

nent

s

minus1

minus25

minus2

minus15

minus05



0 1 2 3 4 5 6 7 8 9 10

0

02

b

Lyap

unov

expo

nent

s

minus08

minus06

minus04

minus02


0

05

1

15

2

25

3

35

4

x

b


(a)

0

05

1

15

2

25

3

35

4

45

x

b

2 4 6 81 3 5 7

(b)



01 015 02 025 03

0

05

a

Lyap

unov

expo

nent

s

minus05

minus1

minus15

minus2

(a)

4 45 5 55 6 65 7 75 8 85 9

0

02

04

F

Lyap

unov

expo

nent

s

minus02

minus04

minus06

minus08

minus1

minus12

(b)


0

05

1

15

2

25

3

35

4

45

a

x

minus0501 012 014 016 018 02 022 024 026 028 03

(a)

0

05

1

15

2

25

3

35

F

x

4 45 5 55 6 65 7 75 8 85 9 95

(b)









4 Conclusion




Acknowledgments


References

















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











2)1199092) = 1198662 that are shown in (3)


119889119909

119889119905


119889119910

119889119905

= 119909119910 minus 119910 + 119866

119889119911

119889119905

= 119909119911 minus 119911

(4)



119909119910 minus 119910 + 119866 = 0

119909119911 minus 119911 = 0

(5)


0 119866(1 minus 119909

0) 0) of model (1) where 119909

0satisfies the



0 119866(1 minus 119909

0) 0) in model (1)









0) 0) is given as

119869 = (

minus119886 minus211991000

11991001199090minus 1 0

0 0 1199090minus 1

) (6)


119891 (120582) = (120582 + 1 minus 1199090) (1205822+ (119886 + 1 minus 119909

0) 120582

+ (119886 + 1) 1199090+ 21199102

0)

(7)



1205821= 1199090minus 1

1205822=


0)2

minus (81198662 (1 minus 119909

0)2

)

2

1205823=


0)2

minus (81198662 (1 minus 119909

0)2

)

2

(8)


1lt 0 Since 81198662(1 minus 119909

0)2gt 0 and


0)2= 4(1 minus 119909


that Re(1205822) lt 0 and Re(120582





0lt


2) lt 0 and Re(120582

3) lt



0 119866(1 minus 119909

0) 0) in model (1)



0 119866(1minus119909


a fold bifurcation



0 119866(1minus119909


a Hopf bifurcation



0= 119886 + 1 21198662 gt 1198864 and 119886 lt 0 Let


3

be 1205721+ 1198941205722and 120572


1 1205722 and 120572

3are all


1199091= 119909 minus 119909

0

1199101= 119910 minus 119910

0

1199111= 119911 minus 119911

0

(9)



1= minus 119910

2

1minus 211991001199101minus 1199102

0minus 1199112

1+ 1198861199090minus 1198861199091+ 119886119865

1199101= 11990911199101+ 11990911199100+ 11990901199101+ 11990901199100minus 1199101minus 1199100+ 119866

1= 11990911199111+ 11990901199111minus 1199111

(10)



1205721015840(0) = Re (1205821015840 (119886)) = 1

2

gt 0

1205961015840(0) = Im (1205821015840 (119886)) =

radic41198662minus 1198864

4 (21198662minus 1198864)

= 0

120596 (0) = Im (120582 (119886)) =radic21198662minus 1198864

|119886|

gt 0

1205823lt 0

(11)


1205721= (

1

0

0

) 1205722= (

119886

119870

119866

119886119870

0

) 1205723= (

0

0

1

)

119879 = (1205722 1205721 1205723) = (

119886

119896

1 0

119866

119886119896

0 0

0 0 1

)

(12)


(

119909111991011199111

) = 119879 (

119909211991021199112

) we have

1199092= 1198601199092+ 119861 (13)

where

119860 = (

0 minus119896 0

119896 0 0

0 0 119886

) 1199092= (

1199092

1199102

1199112

)

119861 =(

119886

119896

1199092

2+ 11990921199102+

119886119896

119866

11990901199100minus 1199100+ 119866

minus

1198662+ 1198864

119886211989621199092

2minus

119886

119896

11990921199102minus 1199112

2minus (

1198862

119866

+

2119866

119886119896

)11990901199100minus 1198861199090+

1198862

119866

1199090+ 119886119865 minus 119886

2

119886

119896

11990921199112+ 11991021199112

)

(14)

From (13)

2= minus 119896119910

2+ 119875 (119909

2 1199102 1199112)

1199102= 1198961199092+ 119876 (119909

2 1199102 1199112)

2= 1198861199112+ 119877 (119909

2 1199102 1199112)

(15)

where

119875 (1199092 1199102 1199112) =

119886

119896

1199092

2+ 11990921199102+

119886119896

119866

11990901199100minus 1199100+ 119866

119876 (1199092 1199102 1199112)

= minus

1198662+ 1198864

119886211989621199092

2minus

119886

119896

11990921199102minus 1199112

2

minus (

1198862

119866

+

2119866

119886119896

)11990901199100minus 1198861199090+

1198862

119866

1199090+ 119886119865 minus 119886

2

119877 (1199092 1199102 1199112) =

119886

119896

11990921199112+ 11991021199112

(16)


11989211=

1

4

(

1205971198752

1205971199092

2

+

1205971198752

1205971199102

2

+ 119894 (

1205971198762

1205971199092

2

+

1205971198762

1205971199102

2

))

=

1

2

(

119886

119896

minus

1198662+ 1198864

11988621198962119894)

11989202=

1

4

(

1205971198752

1205971199092

2

minus

1205971198752

1205971199102

2

minus 2

1205971198762

12059711990921205971199102

+ 119894 (

1205971198762

1205971199092

2

minus

1205971198762

1205971199102

2

+ 2

1205971198752

12059711990921205971199102

))

=

1

2

(1 minus

1198662+ 1198864

11988621198962) 119894

11989220=

1

4

(

1205971198752

1205971199092

2

minus

1205971198752

1205971199102

2

+ 2

1205971198762

12059711990921205971199102

+ 119894 (

1205971198762

1205971199092

2

minus

1205971198762

1205971199102

2

minus 2

1205971198752

12059711990921205971199102

))


=

1

2

(minus1 minus

1198662+ 1198864

11988621198962) 119894

11986621=

1

8

(

1205971198753

1205971199093

2

+

1205971198753

12059711990921205971199102

2

+

1205971198763

1205971199092

21205971199102

+

1205971198763

1205971199103

2

+ 119894 (

1205971198763

1205971199093

2

+

1205971198763

12059711990921205971199102

2

minus

1205971198753

1205971199092

21205971199102

minus

1205971198753

1205971199103

2

)) = 0

(17)


ℎ11=

1

4

(

1205971198772

1205971199092

2

+

1205971198772

1205971199102

2

) = 0

ℎ20=

1

4

(

1205971198772

1205971199092

2

minus

1205971198772

1205971199102

2

minus 2

1205971198772

12059711990921205971199102

) = 0

(18)


120582112060111= minusℎ11

(1205821minus 2119896119894) 120601

20= minusℎ20

(19)

Then

12060111= 0

12060120= 0

(20)

Let

119866110=

1

2

(

1205971198752

12059711990921205971199112

+

1205971198762

12059711991021205971199112

+ 119894 (

1205971198752

12059711990921205971199112

minus

1205971198762

12059711991021205971199112

)) = 0

119866101=

1

2

(

1205971198752

12059711990921205971199112

minus

1205971198762

12059711991021205971199112

+ 119894 (

1205971198752

12059711990921205971199112

+

1205971198762

12059711991021205971199112

)) = 0

(21)


11989221= 11986621+ (2119866

11012060111+ 11986610112060120) = 0 (22)

Also let

1198721(0) =

119894

2119896

(1198922011989211minus 2100381610038161003816100381611989211

1003816100381610038161003816

2

minus

1

3

100381610038161003816100381611989202

1003816100381610038161003816

2

) +

1

2

11989221

=

(119898 + 1) 119886

81198962minus

119894

8119896

(

21198862

1198962+

10

3

1198982+

1

3

119898 +

1

3

)

(23)

Then we have

1205832(119886 119887) = minus

Re (1198721(0))

1205721015840(0)

= minus

(119898 + 1) 119886

41198962gt 0 (119886 lt 0)

1205732(119886 119887) = 2Re (119872

1(0)) =

(119898 + 1) 119886

41198962lt 0 (119886 lt 0)

1205912(119886 119887) = minus

Im (1198721(0)) + 120583

2(119886 119887) 120596

1015840(0)

119887

= minus(

1

81198962(

21198862

1198962+

10

3

1198982+

1

3

119898 +

1

3

)

minus

(119898 + 1) 119886

161198962

radic41198662minus 1198864

4 (21198662minus 1198864)

) lt 0

(24)


0= 119886 + 1 and







2(1 0 119866119887) respectively


1







1is

119869 = (

minus119886 minus2119866 0

119866 minus1 0

119887119866 0 minus1

) (25)


119891 (120582) = (120582 + 1) (1205822+ (119886 + 1) 120582 + 119886 + 2119866

2) (26)



1205821= minus1

12058223=


2)

2

(27)





0 12058223



completed












2is

119869 = (


119866

119887

119866 0 119887

minus

119866

119887

119887 0

) (28)


119891 (120582) = 1205823+ 1198861205822+ (1198872+

21198662

1198872)120582 + 119886119887

2minus 21198662 (29)



proof is completed


1and 119870



0 1199100 1199110)




1199090lt 1 +

119886

2

1199092

0+

4 + 2119865 + 21198651198872

3 (1 + 1198872)

1199090+

1 + 2119865

3 (1 + 1198872)

gt 0

1199093

0minus

5119886 + 21198872minus 1198861198872+ 6

2 (1 + 1198872)

1199092

0+

9119886 + 6 + 119886119865 + 31198862minus 1198861198872119865

2 (1 + 1198872)

1199090

minus

(1 + 119886) (2 + 2119886 + 119886119865)

2 (1 + 1198872)

lt 0

(30)


Proof From (2)

1199100= 119892 (119909

0 119887) (1 minus 119909

0) 119866

1199110= 119892 (119909

0 119887) 1199090119887119866

119886 (119865 minus 119909) 119891 (1199090 119887) = 119866

2

(31)

where119891(1199090 119887) = 1minus2119909

0+(1198872+1)1199092

0and119892(119909

0 119887) = 1119891(119909

0 119887)


given as

119869 (0) =(


0) 119866 minus2119892 (119909

0 119887) 119887119909

0119866

119892 (1199090 119887) (1 minus 119909

0minus 11988721199092

0)119866 119909

0minus 1 minus119887119909

0

119887119892 (1199090 119887) 119866 119887119909

01199090minus 1

) (32)


Taking 119866119892(1199090 119887) = 119879 we can get

119891 (120582) = 1205823+ (2 minus 2119909

0+ 119886) 120582

2

+ (2119886 (1 minus 1199090) + 119891 (119909

0 119887) (1 + 2119879

2)) 120582

+ 119891 (1199090 119887) (119886 + 2119879

2(1 minus 119909

0minus 11988721199090))

(33)

Let 119860 = 2 minus 21199090+ 119886 119861 = 2119886(1 minus 119909

0) + 119891(119909

0 119887)(1 + 2119879

2) and

119862 = 119891(1199090 119887)(119886 + 2119879

2(1 minus 119909

0minus 11988721199090))

Note that the 119891(1199090 119887) gt 0 119892(119909

0 119887) gt 0 1198792 gt 0 and

1199090lt 1 +

119886

2

1199092

0+

4 + 2119865 + 21198651198872

3 (1 + 1198872)

1199090+

1 + 2119865

3 (1 + 1198872)

gt 0

1199093

0minus

5119886 + 21198872minus 1198861198872+ 6

2 (1 + 1198872)

1199092

0

+

9119886 + 6 + 119886119865 + 31198862minus 1198861198872119865

2 (1 + 1198872)

1199090

minus

(1 + 119886) (2 + 2119886 + 119886119865)

2 (1 + 1198872)

lt 0

(34)




2119909 + (1198872+ 1)1199092) = 119866


1199090lt 1 +

119886

2

1199092

0+

4 + 2119865 + 21198651198872

3 (1 + 1198872)

1199090+

1 + 2119865

3 (1 + 1198872)

gt 0

1199093

0minus

5119886 + 21198872minus 1198861198872+ 6

2 (1 + 1198872)

1199092

0

+

9119886 + 6 + 119886119865 + 31198862minus 1198861198872119865

2 (1 + 1198872)

1199090

minus

(1 + 119886) (2 + 2119886 + 119886119865)

2 (1 + 1198872)

= 0

12057311199094

0minus 12057321199093

0+ 12057331199092

0+ 12057341199090+ 1205735= 0

(35)


1205731=

41198862(1 minus 119887

4)

1198662

1205732=

41198862

1198662(2 + 2119887

2+ 119886 + 119886119887

2+ 2119865 minus 2119865119887

4)

1205733=

411988621198652(1 minus 119887

4)

1198662

+

81198651198862(2 + 2119887

2+ 119886 + 119886119887

2)

1198662

+ 3 (1198872+ 1)

1205734= 4119886119887


2minus

41198862119865 (2 + 2119887

2+ 119886 + 119886119887

2)

1198662

1205735= 12119886 + 2119886119865 + 2119886119887

2119865 + 4119886

2+ 3

(36)


1199090)+119891(119909

0 119887)(1+2119879

2) and119862 = 119891(119909

0 119887)(119886+2119879

2(1minus1199090minus11988721199090))

Since

1199090lt 1 +

119886

2

1199092

0+

4 + 2119865 + 21198651198872

3 (1 + 1198872)

1199090+

1 + 2119865

3 (1 + 1198872)

gt 0

1199093

0minus

5119886 + 21198872minus 1198861198872+ 6

2 (1 + 1198872)

1199092

0

+

9119886 + 6 + 119886119865 + 31198862minus 1198861198872119865

2 (1 + 1198872)

1199090

minus

(1 + 119886) (2 + 2119886 + 119886119865)

2 (1 + 1198872)

= 0

(37)


0) = 119862(119909

0)119860(119909

0) gt 0 and


11199094

0minus12057321199093

0+12057331199092

0+12057341199090+1205735= 0


0)1198611015840(1199090) where

1205731=

41198862(1 minus 119887

4)

1198662

1205732=

41198862

1198662(2 + 2119887

2+ 119886 + 119886119887

2+ 2119865 minus 2119865119887

4)

1205733=

411988621198652(1 minus 119887

4)

1198662

+

81198651198862(2 + 2119887

2+ 119886 + 119886119887

2)

1198662

+ 3 (1198872+ 1)

1205734= 4119886119887


2

minus

41198862119865 (2 + 2119887

2+ 119886 + 119886119887

2)

1198662

1205735= 12119886 + 2119886119865 + 2119886119887

2119865 + 4119886

2+ 3

(38)



0) =

119862(1199090)119860(119909

0) gt 0 119862(119909

0) gt 0 and 1198621015840(119909

0) = 119860

1015840(1199090)119861(1199090) minus











2(119886 119887) = 17












119894


119894


119894contained


119894is said to

connect1198631119894and1198632

119894 if 119897 cap 1198631

119894= and 119897 cap 1198632

119894=


1198631

119894and 1198632




119895






0 2 4 6

0

2

4

6

8

G

x

C

B

A


minus2

x = 1

(a)

02

46

8

01

23

45

68

ZY

DC

A

BX024

minus6

minus4

minus2

minus2minus1

(b)



119894and1198632

119894 then


119895and is denoted by119891(119863

119894) sub 119863

119895


119894) and (1198631

119895 1198632

119895)


119895holds for



119890119899119905(119891) ge log119898





1and


1

are(0727230483 1444887781) (0666821561 1360099751)

(0843401487 1245386534) (0913104089 1330174564)

(39)


0

2

4

x A

C

B

minus6

minus4

minus2

0 2 4 6G


minus8

x = 1

(a)

0 1 2 3005

115

ZY

X

0 2

005

115

Z

Y

A

B

C

0

2

4

minus6

minus4

minus2

minus2

minus2minus15minus1

minus05

minus15minus1

minus1

minus05

minus3

(b)


005

1

01

23

45

0123

zy

x

minus1minus05

minus4

minus3

minus2

minus1

(a)

005

1

01

23

45

0123

zy

x

minus1minus05

minus4

minus3

minus2

minus1

(b)


0 02 04 06 08 1 12 14

0

02

04

G

Lyap

unov

expo

nent

s

minus12

minus1

minus08

minus06

minus04

minus02



(1029275093 1245386534) (0964219331 1165586035)

(1085037175 1040897756) (1164033457 1130673317)

(40)






1198675(1198631) 997891997888rarr 119863

1 119867

5(1198631) 997891997888rarr 119863

2

1198675(1198632) 997891997888rarr 119863

1 119867

5(1198632) 997891997888rarr 119863

2

(41)




28

26

24

22

2

18

16

14

12


G

x

(a)

26

24

22

2

18

16

12

14

06 07 08 09 1 11 12 13 14 15

G

x

(b)


minus20

24

minus20

24

0

1

2

3

Z

Y

X

minus1

minus4 minus4

(a)

0 200 400 600 800 1000

0102030405060

Pow

er sp

ectr

um

minus40

minus30

minus20

minus10

10 lowast log10(S)

(b)


0 05 1 15 2 25

0

05

1

15

2

25

x

y

minus25

minus2

minus15

minus1

minus05

minus05

D1

D2

(a)

0 02 04 06 08 1 12 14 16

04

06

08

1

12

14

16

18

x

y

07 08 09 1 111

11

12

13

14

15

Zoom in

D1

D2

D2

D1

D11

D11

D12

D12

D21

D21

D22

H5(D1)

H5(D1)

H5(D2)

H5(D2)

H5(D12)

H5(D11)

H5(D22)

H5(D21)

(b)



0 02 04 06 08 1 12 14 16 18 2

0

05

G

Lyap

unov

expo

nent

s

minus3

minus25

minus2

minus15

minus05

minus1


0 02 04 06 08 1 12 14 16 18 2

0

05

G

Lyap

unov

expo

nent

s

minus1

minus25

minus2

minus15

minus05



0 1 2 3 4 5 6 7 8 9 10

0

02

b

Lyap

unov

expo

nent

s

minus08

minus06

minus04

minus02


0

05

1

15

2

25

3

35

4

x

b


(a)

0

05

1

15

2

25

3

35

4

45

x

b

2 4 6 81 3 5 7

(b)



01 015 02 025 03

0

05

a

Lyap

unov

expo

nent

s

minus05

minus1

minus15

minus2

(a)

4 45 5 55 6 65 7 75 8 85 9

0

02

04

F

Lyap

unov

expo

nent

s

minus02

minus04

minus06

minus08

minus1

minus12

(b)


0

05

1

15

2

25

3

35

4

45

a

x

minus0501 012 014 016 018 02 022 024 026 028 03

(a)

0

05

1

15

2

25

3

35

F

x

4 45 5 55 6 65 7 75 8 85 9 95

(b)









4 Conclusion




Acknowledgments


References

















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1= minus 119910

2

1minus 211991001199101minus 1199102

0minus 1199112

1+ 1198861199090minus 1198861199091+ 119886119865

1199101= 11990911199101+ 11990911199100+ 11990901199101+ 11990901199100minus 1199101minus 1199100+ 119866

1= 11990911199111+ 11990901199111minus 1199111

(10)



1205721015840(0) = Re (1205821015840 (119886)) = 1

2

gt 0

1205961015840(0) = Im (1205821015840 (119886)) =

radic41198662minus 1198864

4 (21198662minus 1198864)

= 0

120596 (0) = Im (120582 (119886)) =radic21198662minus 1198864

|119886|

gt 0

1205823lt 0

(11)


1205721= (

1

0

0

) 1205722= (

119886

119870

119866

119886119870

0

) 1205723= (

0

0

1

)

119879 = (1205722 1205721 1205723) = (

119886

119896

1 0

119866

119886119896

0 0

0 0 1

)

(12)


(

119909111991011199111

) = 119879 (

119909211991021199112

) we have

1199092= 1198601199092+ 119861 (13)

where

119860 = (

0 minus119896 0

119896 0 0

0 0 119886

) 1199092= (

1199092

1199102

1199112

)

119861 =(

119886

119896

1199092

2+ 11990921199102+

119886119896

119866

11990901199100minus 1199100+ 119866

minus

1198662+ 1198864

119886211989621199092

2minus

119886

119896

11990921199102minus 1199112

2minus (

1198862

119866

+

2119866

119886119896

)11990901199100minus 1198861199090+

1198862

119866

1199090+ 119886119865 minus 119886

2

119886

119896

11990921199112+ 11991021199112

)

(14)

From (13)

2= minus 119896119910

2+ 119875 (119909

2 1199102 1199112)

1199102= 1198961199092+ 119876 (119909

2 1199102 1199112)

2= 1198861199112+ 119877 (119909

2 1199102 1199112)

(15)

where

119875 (1199092 1199102 1199112) =

119886

119896

1199092

2+ 11990921199102+

119886119896

119866

11990901199100minus 1199100+ 119866

119876 (1199092 1199102 1199112)

= minus

1198662+ 1198864

119886211989621199092

2minus

119886

119896

11990921199102minus 1199112

2

minus (

1198862

119866

+

2119866

119886119896

)11990901199100minus 1198861199090+

1198862

119866

1199090+ 119886119865 minus 119886

2

119877 (1199092 1199102 1199112) =

119886

119896

11990921199112+ 11991021199112

(16)


11989211=

1

4

(

1205971198752

1205971199092

2

+

1205971198752

1205971199102

2

+ 119894 (

1205971198762

1205971199092

2

+

1205971198762

1205971199102

2

))

=

1

2

(

119886

119896

minus

1198662+ 1198864

11988621198962119894)

11989202=

1

4

(

1205971198752

1205971199092

2

minus

1205971198752

1205971199102

2

minus 2

1205971198762

12059711990921205971199102

+ 119894 (

1205971198762

1205971199092

2

minus

1205971198762

1205971199102

2

+ 2

1205971198752

12059711990921205971199102

))

=

1

2

(1 minus

1198662+ 1198864

11988621198962) 119894

11989220=

1

4

(

1205971198752

1205971199092

2

minus

1205971198752

1205971199102

2

+ 2

1205971198762

12059711990921205971199102

+ 119894 (

1205971198762

1205971199092

2

minus

1205971198762

1205971199102

2

minus 2

1205971198752

12059711990921205971199102

))


=

1

2

(minus1 minus

1198662+ 1198864

11988621198962) 119894

11986621=

1

8

(

1205971198753

1205971199093

2

+

1205971198753

12059711990921205971199102

2

+

1205971198763

1205971199092

21205971199102

+

1205971198763

1205971199103

2

+ 119894 (

1205971198763

1205971199093

2

+

1205971198763

12059711990921205971199102

2

minus

1205971198753

1205971199092

21205971199102

minus

1205971198753

1205971199103

2

)) = 0

(17)


ℎ11=

1

4

(

1205971198772

1205971199092

2

+

1205971198772

1205971199102

2

) = 0

ℎ20=

1

4

(

1205971198772

1205971199092

2

minus

1205971198772

1205971199102

2

minus 2

1205971198772

12059711990921205971199102

) = 0

(18)


120582112060111= minusℎ11

(1205821minus 2119896119894) 120601

20= minusℎ20

(19)

Then

12060111= 0

12060120= 0

(20)

Let

119866110=

1

2

(

1205971198752

12059711990921205971199112

+

1205971198762

12059711991021205971199112

+ 119894 (

1205971198752

12059711990921205971199112

minus

1205971198762

12059711991021205971199112

)) = 0

119866101=

1

2

(

1205971198752

12059711990921205971199112

minus

1205971198762

12059711991021205971199112

+ 119894 (

1205971198752

12059711990921205971199112

+

1205971198762

12059711991021205971199112

)) = 0

(21)


11989221= 11986621+ (2119866

11012060111+ 11986610112060120) = 0 (22)

Also let

1198721(0) =

119894

2119896

(1198922011989211minus 2100381610038161003816100381611989211

1003816100381610038161003816

2

minus

1

3

100381610038161003816100381611989202

1003816100381610038161003816

2

) +

1

2

11989221

=

(119898 + 1) 119886

81198962minus

119894

8119896

(

21198862

1198962+

10

3

1198982+

1

3

119898 +

1

3

)

(23)

Then we have

1205832(119886 119887) = minus

Re (1198721(0))

1205721015840(0)

= minus

(119898 + 1) 119886

41198962gt 0 (119886 lt 0)

1205732(119886 119887) = 2Re (119872

1(0)) =

(119898 + 1) 119886

41198962lt 0 (119886 lt 0)

1205912(119886 119887) = minus

Im (1198721(0)) + 120583

2(119886 119887) 120596

1015840(0)

119887

= minus(

1

81198962(

21198862

1198962+

10

3

1198982+

1

3

119898 +

1

3

)

minus

(119898 + 1) 119886

161198962

radic41198662minus 1198864

4 (21198662minus 1198864)

) lt 0

(24)


0= 119886 + 1 and







2(1 0 119866119887) respectively


1







1is

119869 = (

minus119886 minus2119866 0

119866 minus1 0

119887119866 0 minus1

) (25)


119891 (120582) = (120582 + 1) (1205822+ (119886 + 1) 120582 + 119886 + 2119866

2) (26)



1205821= minus1

12058223=


2)

2

(27)





0 12058223



completed












2is

119869 = (


119866

119887

119866 0 119887

minus

119866

119887

119887 0

) (28)


119891 (120582) = 1205823+ 1198861205822+ (1198872+

21198662

1198872)120582 + 119886119887

2minus 21198662 (29)



proof is completed


1and 119870



0 1199100 1199110)




1199090lt 1 +

119886

2

1199092

0+

4 + 2119865 + 21198651198872

3 (1 + 1198872)

1199090+

1 + 2119865

3 (1 + 1198872)

gt 0

1199093

0minus

5119886 + 21198872minus 1198861198872+ 6

2 (1 + 1198872)

1199092

0+

9119886 + 6 + 119886119865 + 31198862minus 1198861198872119865

2 (1 + 1198872)

1199090

minus

(1 + 119886) (2 + 2119886 + 119886119865)

2 (1 + 1198872)

lt 0

(30)


Proof From (2)

1199100= 119892 (119909

0 119887) (1 minus 119909

0) 119866

1199110= 119892 (119909

0 119887) 1199090119887119866

119886 (119865 minus 119909) 119891 (1199090 119887) = 119866

2

(31)

where119891(1199090 119887) = 1minus2119909

0+(1198872+1)1199092

0and119892(119909

0 119887) = 1119891(119909

0 119887)


given as

119869 (0) =(


0) 119866 minus2119892 (119909

0 119887) 119887119909

0119866

119892 (1199090 119887) (1 minus 119909

0minus 11988721199092

0)119866 119909

0minus 1 minus119887119909

0

119887119892 (1199090 119887) 119866 119887119909

01199090minus 1

) (32)


Taking 119866119892(1199090 119887) = 119879 we can get

119891 (120582) = 1205823+ (2 minus 2119909

0+ 119886) 120582

2

+ (2119886 (1 minus 1199090) + 119891 (119909

0 119887) (1 + 2119879

2)) 120582

+ 119891 (1199090 119887) (119886 + 2119879

2(1 minus 119909

0minus 11988721199090))

(33)

Let 119860 = 2 minus 21199090+ 119886 119861 = 2119886(1 minus 119909

0) + 119891(119909

0 119887)(1 + 2119879

2) and

119862 = 119891(1199090 119887)(119886 + 2119879

2(1 minus 119909

0minus 11988721199090))

Note that the 119891(1199090 119887) gt 0 119892(119909

0 119887) gt 0 1198792 gt 0 and

1199090lt 1 +

119886

2

1199092

0+

4 + 2119865 + 21198651198872

3 (1 + 1198872)

1199090+

1 + 2119865

3 (1 + 1198872)

gt 0

1199093

0minus

5119886 + 21198872minus 1198861198872+ 6

2 (1 + 1198872)

1199092

0

+

9119886 + 6 + 119886119865 + 31198862minus 1198861198872119865

2 (1 + 1198872)

1199090

minus

(1 + 119886) (2 + 2119886 + 119886119865)

2 (1 + 1198872)

lt 0

(34)




2119909 + (1198872+ 1)1199092) = 119866


1199090lt 1 +

119886

2

1199092

0+

4 + 2119865 + 21198651198872

3 (1 + 1198872)

1199090+

1 + 2119865

3 (1 + 1198872)

gt 0

1199093

0minus

5119886 + 21198872minus 1198861198872+ 6

2 (1 + 1198872)

1199092

0

+

9119886 + 6 + 119886119865 + 31198862minus 1198861198872119865

2 (1 + 1198872)

1199090

minus

(1 + 119886) (2 + 2119886 + 119886119865)

2 (1 + 1198872)

= 0

12057311199094

0minus 12057321199093

0+ 12057331199092

0+ 12057341199090+ 1205735= 0

(35)


1205731=

41198862(1 minus 119887

4)

1198662

1205732=

41198862

1198662(2 + 2119887

2+ 119886 + 119886119887

2+ 2119865 minus 2119865119887

4)

1205733=

411988621198652(1 minus 119887

4)

1198662

+

81198651198862(2 + 2119887

2+ 119886 + 119886119887

2)

1198662

+ 3 (1198872+ 1)

1205734= 4119886119887


2minus

41198862119865 (2 + 2119887

2+ 119886 + 119886119887

2)

1198662

1205735= 12119886 + 2119886119865 + 2119886119887

2119865 + 4119886

2+ 3

(36)


1199090)+119891(119909

0 119887)(1+2119879

2) and119862 = 119891(119909

0 119887)(119886+2119879

2(1minus1199090minus11988721199090))

Since

1199090lt 1 +

119886

2

1199092

0+

4 + 2119865 + 21198651198872

3 (1 + 1198872)

1199090+

1 + 2119865

3 (1 + 1198872)

gt 0

1199093

0minus

5119886 + 21198872minus 1198861198872+ 6

2 (1 + 1198872)

1199092

0

+

9119886 + 6 + 119886119865 + 31198862minus 1198861198872119865

2 (1 + 1198872)

1199090

minus

(1 + 119886) (2 + 2119886 + 119886119865)

2 (1 + 1198872)

= 0

(37)


0) = 119862(119909

0)119860(119909

0) gt 0 and


11199094

0minus12057321199093

0+12057331199092

0+12057341199090+1205735= 0


0)1198611015840(1199090) where

1205731=

41198862(1 minus 119887

4)

1198662

1205732=

41198862

1198662(2 + 2119887

2+ 119886 + 119886119887

2+ 2119865 minus 2119865119887

4)

1205733=

411988621198652(1 minus 119887

4)

1198662

+

81198651198862(2 + 2119887

2+ 119886 + 119886119887

2)

1198662

+ 3 (1198872+ 1)

1205734= 4119886119887


2

minus

41198862119865 (2 + 2119887

2+ 119886 + 119886119887

2)

1198662

1205735= 12119886 + 2119886119865 + 2119886119887

2119865 + 4119886

2+ 3

(38)



0) =

119862(1199090)119860(119909

0) gt 0 119862(119909

0) gt 0 and 1198621015840(119909

0) = 119860

1015840(1199090)119861(1199090) minus











2(119886 119887) = 17












119894


119894


119894contained


119894is said to

connect1198631119894and1198632

119894 if 119897 cap 1198631

119894= and 119897 cap 1198632

119894=


1198631

119894and 1198632




119895






0 2 4 6

0

2

4

6

8

G

x

C

B

A


minus2

x = 1

(a)

02

46

8

01

23

45

68

ZY

DC

A

BX024

minus6

minus4

minus2

minus2minus1

(b)



119894and1198632

119894 then


119895and is denoted by119891(119863

119894) sub 119863

119895


119894) and (1198631

119895 1198632

119895)


119895holds for



119890119899119905(119891) ge log119898





1and


1

are(0727230483 1444887781) (0666821561 1360099751)

(0843401487 1245386534) (0913104089 1330174564)

(39)


0

2

4

x A

C

B

minus6

minus4

minus2

0 2 4 6G


minus8

x = 1

(a)

0 1 2 3005

115

ZY

X

0 2

005

115

Z

Y

A

B

C

0

2

4

minus6

minus4

minus2

minus2

minus2minus15minus1

minus05

minus15minus1

minus1

minus05

minus3

(b)


005

1

01

23

45

0123

zy

x

minus1minus05

minus4

minus3

minus2

minus1

(a)

005

1

01

23

45

0123

zy

x

minus1minus05

minus4

minus3

minus2

minus1

(b)


0 02 04 06 08 1 12 14

0

02

04

G

Lyap

unov

expo

nent

s

minus12

minus1

minus08

minus06

minus04

minus02



(1029275093 1245386534) (0964219331 1165586035)

(1085037175 1040897756) (1164033457 1130673317)

(40)






1198675(1198631) 997891997888rarr 119863

1 119867

5(1198631) 997891997888rarr 119863

2

1198675(1198632) 997891997888rarr 119863

1 119867

5(1198632) 997891997888rarr 119863

2

(41)




28

26

24

22

2

18

16

14

12


G

x

(a)

26

24

22

2

18

16

12

14

06 07 08 09 1 11 12 13 14 15

G

x

(b)


minus20

24

minus20

24

0

1

2

3

Z

Y

X

minus1

minus4 minus4

(a)

0 200 400 600 800 1000

0102030405060

Pow

er sp

ectr

um

minus40

minus30

minus20

minus10

10 lowast log10(S)

(b)


0 05 1 15 2 25

0

05

1

15

2

25

x

y

minus25

minus2

minus15

minus1

minus05

minus05

D1

D2

(a)

0 02 04 06 08 1 12 14 16

04

06

08

1

12

14

16

18

x

y

07 08 09 1 111

11

12

13

14

15

Zoom in

D1

D2

D2

D1

D11

D11

D12

D12

D21

D21

D22

H5(D1)

H5(D1)

H5(D2)

H5(D2)

H5(D12)

H5(D11)

H5(D22)

H5(D21)

(b)



0 02 04 06 08 1 12 14 16 18 2

0

05

G

Lyap

unov

expo

nent

s

minus3

minus25

minus2

minus15

minus05

minus1


0 02 04 06 08 1 12 14 16 18 2

0

05

G

Lyap

unov

expo

nent

s

minus1

minus25

minus2

minus15

minus05



0 1 2 3 4 5 6 7 8 9 10

0

02

b

Lyap

unov

expo

nent

s

minus08

minus06

minus04

minus02


0

05

1

15

2

25

3

35

4

x

b


(a)

0

05

1

15

2

25

3

35

4

45

x

b

2 4 6 81 3 5 7

(b)



01 015 02 025 03

0

05

a

Lyap

unov

expo

nent

s

minus05

minus1

minus15

minus2

(a)

4 45 5 55 6 65 7 75 8 85 9

0

02

04

F

Lyap

unov

expo

nent

s

minus02

minus04

minus06

minus08

minus1

minus12

(b)


0

05

1

15

2

25

3

35

4

45

a

x

minus0501 012 014 016 018 02 022 024 026 028 03

(a)

0

05

1

15

2

25

3

35

F

x

4 45 5 55 6 65 7 75 8 85 9 95

(b)









4 Conclusion




Acknowledgments


References

















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










=

1

2

(minus1 minus

1198662+ 1198864

11988621198962) 119894

11986621=

1

8

(

1205971198753

1205971199093

2

+

1205971198753

12059711990921205971199102

2

+

1205971198763

1205971199092

21205971199102

+

1205971198763

1205971199103

2

+ 119894 (

1205971198763

1205971199093

2

+

1205971198763

12059711990921205971199102

2

minus

1205971198753

1205971199092

21205971199102

minus

1205971198753

1205971199103

2

)) = 0

(17)


ℎ11=

1

4

(

1205971198772

1205971199092

2

+

1205971198772

1205971199102

2

) = 0

ℎ20=

1

4

(

1205971198772

1205971199092

2

minus

1205971198772

1205971199102

2

minus 2

1205971198772

12059711990921205971199102

) = 0

(18)


120582112060111= minusℎ11

(1205821minus 2119896119894) 120601

20= minusℎ20

(19)

Then

12060111= 0

12060120= 0

(20)

Let

119866110=

1

2

(

1205971198752

12059711990921205971199112

+

1205971198762

12059711991021205971199112

+ 119894 (

1205971198752

12059711990921205971199112

minus

1205971198762

12059711991021205971199112

)) = 0

119866101=

1

2

(

1205971198752

12059711990921205971199112

minus

1205971198762

12059711991021205971199112

+ 119894 (

1205971198752

12059711990921205971199112

+

1205971198762

12059711991021205971199112

)) = 0

(21)


11989221= 11986621+ (2119866

11012060111+ 11986610112060120) = 0 (22)

Also let

1198721(0) =

119894

2119896

(1198922011989211minus 2100381610038161003816100381611989211

1003816100381610038161003816

2

minus

1

3

100381610038161003816100381611989202

1003816100381610038161003816

2

) +

1

2

11989221

=

(119898 + 1) 119886

81198962minus

119894

8119896

(

21198862

1198962+

10

3

1198982+

1

3

119898 +

1

3

)

(23)

Then we have

1205832(119886 119887) = minus

Re (1198721(0))

1205721015840(0)

= minus

(119898 + 1) 119886

41198962gt 0 (119886 lt 0)

1205732(119886 119887) = 2Re (119872

1(0)) =

(119898 + 1) 119886

41198962lt 0 (119886 lt 0)

1205912(119886 119887) = minus

Im (1198721(0)) + 120583

2(119886 119887) 120596

1015840(0)

119887

= minus(

1

81198962(

21198862

1198962+

10

3

1198982+

1

3

119898 +

1

3

)

minus

(119898 + 1) 119886

161198962

radic41198662minus 1198864

4 (21198662minus 1198864)

) lt 0

(24)


0= 119886 + 1 and







2(1 0 119866119887) respectively


1







1is

119869 = (

minus119886 minus2119866 0

119866 minus1 0

119887119866 0 minus1

) (25)


119891 (120582) = (120582 + 1) (1205822+ (119886 + 1) 120582 + 119886 + 2119866

2) (26)



1205821= minus1

12058223=


2)

2

(27)





0 12058223



completed












2is

119869 = (


119866

119887

119866 0 119887

minus

119866

119887

119887 0

) (28)


119891 (120582) = 1205823+ 1198861205822+ (1198872+

21198662

1198872)120582 + 119886119887

2minus 21198662 (29)



proof is completed


1and 119870



0 1199100 1199110)




1199090lt 1 +

119886

2

1199092

0+

4 + 2119865 + 21198651198872

3 (1 + 1198872)

1199090+

1 + 2119865

3 (1 + 1198872)

gt 0

1199093

0minus

5119886 + 21198872minus 1198861198872+ 6

2 (1 + 1198872)

1199092

0+

9119886 + 6 + 119886119865 + 31198862minus 1198861198872119865

2 (1 + 1198872)

1199090

minus

(1 + 119886) (2 + 2119886 + 119886119865)

2 (1 + 1198872)

lt 0

(30)


Proof From (2)

1199100= 119892 (119909

0 119887) (1 minus 119909

0) 119866

1199110= 119892 (119909

0 119887) 1199090119887119866

119886 (119865 minus 119909) 119891 (1199090 119887) = 119866

2

(31)

where119891(1199090 119887) = 1minus2119909

0+(1198872+1)1199092

0and119892(119909

0 119887) = 1119891(119909

0 119887)


given as

119869 (0) =(


0) 119866 minus2119892 (119909

0 119887) 119887119909

0119866

119892 (1199090 119887) (1 minus 119909

0minus 11988721199092

0)119866 119909

0minus 1 minus119887119909

0

119887119892 (1199090 119887) 119866 119887119909

01199090minus 1

) (32)


Taking 119866119892(1199090 119887) = 119879 we can get

119891 (120582) = 1205823+ (2 minus 2119909

0+ 119886) 120582

2

+ (2119886 (1 minus 1199090) + 119891 (119909

0 119887) (1 + 2119879

2)) 120582

+ 119891 (1199090 119887) (119886 + 2119879

2(1 minus 119909

0minus 11988721199090))

(33)

Let 119860 = 2 minus 21199090+ 119886 119861 = 2119886(1 minus 119909

0) + 119891(119909

0 119887)(1 + 2119879

2) and

119862 = 119891(1199090 119887)(119886 + 2119879

2(1 minus 119909

0minus 11988721199090))

Note that the 119891(1199090 119887) gt 0 119892(119909

0 119887) gt 0 1198792 gt 0 and

1199090lt 1 +

119886

2

1199092

0+

4 + 2119865 + 21198651198872

3 (1 + 1198872)

1199090+

1 + 2119865

3 (1 + 1198872)

gt 0

1199093

0minus

5119886 + 21198872minus 1198861198872+ 6

2 (1 + 1198872)

1199092

0

+

9119886 + 6 + 119886119865 + 31198862minus 1198861198872119865

2 (1 + 1198872)

1199090

minus

(1 + 119886) (2 + 2119886 + 119886119865)

2 (1 + 1198872)

lt 0

(34)




2119909 + (1198872+ 1)1199092) = 119866


1199090lt 1 +

119886

2

1199092

0+

4 + 2119865 + 21198651198872

3 (1 + 1198872)

1199090+

1 + 2119865

3 (1 + 1198872)

gt 0

1199093

0minus

5119886 + 21198872minus 1198861198872+ 6

2 (1 + 1198872)

1199092

0

+

9119886 + 6 + 119886119865 + 31198862minus 1198861198872119865

2 (1 + 1198872)

1199090

minus

(1 + 119886) (2 + 2119886 + 119886119865)

2 (1 + 1198872)

= 0

12057311199094

0minus 12057321199093

0+ 12057331199092

0+ 12057341199090+ 1205735= 0

(35)


1205731=

41198862(1 minus 119887

4)

1198662

1205732=

41198862

1198662(2 + 2119887

2+ 119886 + 119886119887

2+ 2119865 minus 2119865119887

4)

1205733=

411988621198652(1 minus 119887

4)

1198662

+

81198651198862(2 + 2119887

2+ 119886 + 119886119887

2)

1198662

+ 3 (1198872+ 1)

1205734= 4119886119887


2minus

41198862119865 (2 + 2119887

2+ 119886 + 119886119887

2)

1198662

1205735= 12119886 + 2119886119865 + 2119886119887

2119865 + 4119886

2+ 3

(36)


1199090)+119891(119909

0 119887)(1+2119879

2) and119862 = 119891(119909

0 119887)(119886+2119879

2(1minus1199090minus11988721199090))

Since

1199090lt 1 +

119886

2

1199092

0+

4 + 2119865 + 21198651198872

3 (1 + 1198872)

1199090+

1 + 2119865

3 (1 + 1198872)

gt 0

1199093

0minus

5119886 + 21198872minus 1198861198872+ 6

2 (1 + 1198872)

1199092

0

+

9119886 + 6 + 119886119865 + 31198862minus 1198861198872119865

2 (1 + 1198872)

1199090

minus

(1 + 119886) (2 + 2119886 + 119886119865)

2 (1 + 1198872)

= 0

(37)


0) = 119862(119909

0)119860(119909

0) gt 0 and


11199094

0minus12057321199093

0+12057331199092

0+12057341199090+1205735= 0


0)1198611015840(1199090) where

1205731=

41198862(1 minus 119887

4)

1198662

1205732=

41198862

1198662(2 + 2119887

2+ 119886 + 119886119887

2+ 2119865 minus 2119865119887

4)

1205733=

411988621198652(1 minus 119887

4)

1198662

+

81198651198862(2 + 2119887

2+ 119886 + 119886119887

2)

1198662

+ 3 (1198872+ 1)

1205734= 4119886119887


2

minus

41198862119865 (2 + 2119887

2+ 119886 + 119886119887

2)

1198662

1205735= 12119886 + 2119886119865 + 2119886119887

2119865 + 4119886

2+ 3

(38)



0) =

119862(1199090)119860(119909

0) gt 0 119862(119909

0) gt 0 and 1198621015840(119909

0) = 119860

1015840(1199090)119861(1199090) minus











2(119886 119887) = 17












119894


119894


119894contained


119894is said to

connect1198631119894and1198632

119894 if 119897 cap 1198631

119894= and 119897 cap 1198632

119894=


1198631

119894and 1198632




119895






0 2 4 6

0

2

4

6

8

G

x

C

B

A


minus2

x = 1

(a)

02

46

8

01

23

45

68

ZY

DC

A

BX024

minus6

minus4

minus2

minus2minus1

(b)



119894and1198632

119894 then


119895and is denoted by119891(119863

119894) sub 119863

119895


119894) and (1198631

119895 1198632

119895)


119895holds for



119890119899119905(119891) ge log119898





1and


1

are(0727230483 1444887781) (0666821561 1360099751)

(0843401487 1245386534) (0913104089 1330174564)

(39)


0

2

4

x A

C

B

minus6

minus4

minus2

0 2 4 6G


minus8

x = 1

(a)

0 1 2 3005

115

ZY

X

0 2

005

115

Z

Y

A

B

C

0

2

4

minus6

minus4

minus2

minus2

minus2minus15minus1

minus05

minus15minus1

minus1

minus05

minus3

(b)


005

1

01

23

45

0123

zy

x

minus1minus05

minus4

minus3

minus2

minus1

(a)

005

1

01

23

45

0123

zy

x

minus1minus05

minus4

minus3

minus2

minus1

(b)


0 02 04 06 08 1 12 14

0

02

04

G

Lyap

unov

expo

nent

s

minus12

minus1

minus08

minus06

minus04

minus02



(1029275093 1245386534) (0964219331 1165586035)

(1085037175 1040897756) (1164033457 1130673317)

(40)






1198675(1198631) 997891997888rarr 119863

1 119867

5(1198631) 997891997888rarr 119863

2

1198675(1198632) 997891997888rarr 119863

1 119867

5(1198632) 997891997888rarr 119863

2

(41)




28

26

24

22

2

18

16

14

12


G

x

(a)

26

24

22

2

18

16

12

14

06 07 08 09 1 11 12 13 14 15

G

x

(b)


minus20

24

minus20

24

0

1

2

3

Z

Y

X

minus1

minus4 minus4

(a)

0 200 400 600 800 1000

0102030405060

Pow

er sp

ectr

um

minus40

minus30

minus20

minus10

10 lowast log10(S)

(b)


0 05 1 15 2 25

0

05

1

15

2

25

x

y

minus25

minus2

minus15

minus1

minus05

minus05

D1

D2

(a)

0 02 04 06 08 1 12 14 16

04

06

08

1

12

14

16

18

x

y

07 08 09 1 111

11

12

13

14

15

Zoom in

D1

D2

D2

D1

D11

D11

D12

D12

D21

D21

D22

H5(D1)

H5(D1)

H5(D2)

H5(D2)

H5(D12)

H5(D11)

H5(D22)

H5(D21)

(b)



0 02 04 06 08 1 12 14 16 18 2

0

05

G

Lyap

unov

expo

nent

s

minus3

minus25

minus2

minus15

minus05

minus1


0 02 04 06 08 1 12 14 16 18 2

0

05

G

Lyap

unov

expo

nent

s

minus1

minus25

minus2

minus15

minus05



0 1 2 3 4 5 6 7 8 9 10

0

02

b

Lyap

unov

expo

nent

s

minus08

minus06

minus04

minus02


0

05

1

15

2

25

3

35

4

x

b


(a)

0

05

1

15

2

25

3

35

4

45

x

b

2 4 6 81 3 5 7

(b)



01 015 02 025 03

0

05

a

Lyap

unov

expo

nent

s

minus05

minus1

minus15

minus2

(a)

4 45 5 55 6 65 7 75 8 85 9

0

02

04

F

Lyap

unov

expo

nent

s

minus02

minus04

minus06

minus08

minus1

minus12

(b)


0

05

1

15

2

25

3

35

4

45

a

x

minus0501 012 014 016 018 02 022 024 026 028 03

(a)

0

05

1

15

2

25

3

35

F

x

4 45 5 55 6 65 7 75 8 85 9 95

(b)









4 Conclusion




Acknowledgments


References

















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1205821= minus1

12058223=


2)

2

(27)





0 12058223



completed












2is

119869 = (


119866

119887

119866 0 119887

minus

119866

119887

119887 0

) (28)


119891 (120582) = 1205823+ 1198861205822+ (1198872+

21198662

1198872)120582 + 119886119887

2minus 21198662 (29)



proof is completed


1and 119870



0 1199100 1199110)




1199090lt 1 +

119886

2

1199092

0+

4 + 2119865 + 21198651198872

3 (1 + 1198872)

1199090+

1 + 2119865

3 (1 + 1198872)

gt 0

1199093

0minus

5119886 + 21198872minus 1198861198872+ 6

2 (1 + 1198872)

1199092

0+

9119886 + 6 + 119886119865 + 31198862minus 1198861198872119865

2 (1 + 1198872)

1199090

minus

(1 + 119886) (2 + 2119886 + 119886119865)

2 (1 + 1198872)

lt 0

(30)


Proof From (2)

1199100= 119892 (119909

0 119887) (1 minus 119909

0) 119866

1199110= 119892 (119909

0 119887) 1199090119887119866

119886 (119865 minus 119909) 119891 (1199090 119887) = 119866

2

(31)

where119891(1199090 119887) = 1minus2119909

0+(1198872+1)1199092

0and119892(119909

0 119887) = 1119891(119909

0 119887)


given as

119869 (0) =(


0) 119866 minus2119892 (119909

0 119887) 119887119909

0119866

119892 (1199090 119887) (1 minus 119909

0minus 11988721199092

0)119866 119909

0minus 1 minus119887119909

0

119887119892 (1199090 119887) 119866 119887119909

01199090minus 1

) (32)


Taking 119866119892(1199090 119887) = 119879 we can get

119891 (120582) = 1205823+ (2 minus 2119909

0+ 119886) 120582

2

+ (2119886 (1 minus 1199090) + 119891 (119909

0 119887) (1 + 2119879

2)) 120582

+ 119891 (1199090 119887) (119886 + 2119879

2(1 minus 119909

0minus 11988721199090))

(33)

Let 119860 = 2 minus 21199090+ 119886 119861 = 2119886(1 minus 119909

0) + 119891(119909

0 119887)(1 + 2119879

2) and

119862 = 119891(1199090 119887)(119886 + 2119879

2(1 minus 119909

0minus 11988721199090))

Note that the 119891(1199090 119887) gt 0 119892(119909

0 119887) gt 0 1198792 gt 0 and

1199090lt 1 +

119886

2

1199092

0+

4 + 2119865 + 21198651198872

3 (1 + 1198872)

1199090+

1 + 2119865

3 (1 + 1198872)

gt 0

1199093

0minus

5119886 + 21198872minus 1198861198872+ 6

2 (1 + 1198872)

1199092

0

+

9119886 + 6 + 119886119865 + 31198862minus 1198861198872119865

2 (1 + 1198872)

1199090

minus

(1 + 119886) (2 + 2119886 + 119886119865)

2 (1 + 1198872)

lt 0

(34)




2119909 + (1198872+ 1)1199092) = 119866


1199090lt 1 +

119886

2

1199092

0+

4 + 2119865 + 21198651198872

3 (1 + 1198872)

1199090+

1 + 2119865

3 (1 + 1198872)

gt 0

1199093

0minus

5119886 + 21198872minus 1198861198872+ 6

2 (1 + 1198872)

1199092

0

+

9119886 + 6 + 119886119865 + 31198862minus 1198861198872119865

2 (1 + 1198872)

1199090

minus

(1 + 119886) (2 + 2119886 + 119886119865)

2 (1 + 1198872)

= 0

12057311199094

0minus 12057321199093

0+ 12057331199092

0+ 12057341199090+ 1205735= 0

(35)


1205731=

41198862(1 minus 119887

4)

1198662

1205732=

41198862

1198662(2 + 2119887

2+ 119886 + 119886119887

2+ 2119865 minus 2119865119887

4)

1205733=

411988621198652(1 minus 119887

4)

1198662

+

81198651198862(2 + 2119887

2+ 119886 + 119886119887

2)

1198662

+ 3 (1198872+ 1)

1205734= 4119886119887


2minus

41198862119865 (2 + 2119887

2+ 119886 + 119886119887

2)

1198662

1205735= 12119886 + 2119886119865 + 2119886119887

2119865 + 4119886

2+ 3

(36)


1199090)+119891(119909

0 119887)(1+2119879

2) and119862 = 119891(119909

0 119887)(119886+2119879

2(1minus1199090minus11988721199090))

Since

1199090lt 1 +

119886

2

1199092

0+

4 + 2119865 + 21198651198872

3 (1 + 1198872)

1199090+

1 + 2119865

3 (1 + 1198872)

gt 0

1199093

0minus

5119886 + 21198872minus 1198861198872+ 6

2 (1 + 1198872)

1199092

0

+

9119886 + 6 + 119886119865 + 31198862minus 1198861198872119865

2 (1 + 1198872)

1199090

minus

(1 + 119886) (2 + 2119886 + 119886119865)

2 (1 + 1198872)

= 0

(37)


0) = 119862(119909

0)119860(119909

0) gt 0 and


11199094

0minus12057321199093

0+12057331199092

0+12057341199090+1205735= 0


0)1198611015840(1199090) where

1205731=

41198862(1 minus 119887

4)

1198662

1205732=

41198862

1198662(2 + 2119887

2+ 119886 + 119886119887

2+ 2119865 minus 2119865119887

4)

1205733=

411988621198652(1 minus 119887

4)

1198662

+

81198651198862(2 + 2119887

2+ 119886 + 119886119887

2)

1198662

+ 3 (1198872+ 1)

1205734= 4119886119887


2

minus

41198862119865 (2 + 2119887

2+ 119886 + 119886119887

2)

1198662

1205735= 12119886 + 2119886119865 + 2119886119887

2119865 + 4119886

2+ 3

(38)



0) =

119862(1199090)119860(119909

0) gt 0 119862(119909

0) gt 0 and 1198621015840(119909

0) = 119860

1015840(1199090)119861(1199090) minus











2(119886 119887) = 17












119894


119894


119894contained


119894is said to

connect1198631119894and1198632

119894 if 119897 cap 1198631

119894= and 119897 cap 1198632

119894=


1198631

119894and 1198632




119895






0 2 4 6

0

2

4

6

8

G

x

C

B

A


minus2

x = 1

(a)

02

46

8

01

23

45

68

ZY

DC

A

BX024

minus6

minus4

minus2

minus2minus1

(b)



119894and1198632

119894 then


119895and is denoted by119891(119863

119894) sub 119863

119895


119894) and (1198631

119895 1198632

119895)


119895holds for



119890119899119905(119891) ge log119898





1and


1

are(0727230483 1444887781) (0666821561 1360099751)

(0843401487 1245386534) (0913104089 1330174564)

(39)


0

2

4

x A

C

B

minus6

minus4

minus2

0 2 4 6G


minus8

x = 1

(a)

0 1 2 3005

115

ZY

X

0 2

005

115

Z

Y

A

B

C

0

2

4

minus6

minus4

minus2

minus2

minus2minus15minus1

minus05

minus15minus1

minus1

minus05

minus3

(b)


005

1

01

23

45

0123

zy

x

minus1minus05

minus4

minus3

minus2

minus1

(a)

005

1

01

23

45

0123

zy

x

minus1minus05

minus4

minus3

minus2

minus1

(b)


0 02 04 06 08 1 12 14

0

02

04

G

Lyap

unov

expo

nent

s

minus12

minus1

minus08

minus06

minus04

minus02



(1029275093 1245386534) (0964219331 1165586035)

(1085037175 1040897756) (1164033457 1130673317)

(40)






1198675(1198631) 997891997888rarr 119863

1 119867

5(1198631) 997891997888rarr 119863

2

1198675(1198632) 997891997888rarr 119863

1 119867

5(1198632) 997891997888rarr 119863

2

(41)




28

26

24

22

2

18

16

14

12


G

x

(a)

26

24

22

2

18

16

12

14

06 07 08 09 1 11 12 13 14 15

G

x

(b)


minus20

24

minus20

24

0

1

2

3

Z

Y

X

minus1

minus4 minus4

(a)

0 200 400 600 800 1000

0102030405060

Pow

er sp

ectr

um

minus40

minus30

minus20

minus10

10 lowast log10(S)

(b)


0 05 1 15 2 25

0

05

1

15

2

25

x

y

minus25

minus2

minus15

minus1

minus05

minus05

D1

D2

(a)

0 02 04 06 08 1 12 14 16

04

06

08

1

12

14

16

18

x

y

07 08 09 1 111

11

12

13

14

15

Zoom in

D1

D2

D2

D1

D11

D11

D12

D12

D21

D21

D22

H5(D1)

H5(D1)

H5(D2)

H5(D2)

H5(D12)

H5(D11)

H5(D22)

H5(D21)

(b)



0 02 04 06 08 1 12 14 16 18 2

0

05

G

Lyap

unov

expo

nent

s

minus3

minus25

minus2

minus15

minus05

minus1


0 02 04 06 08 1 12 14 16 18 2

0

05

G

Lyap

unov

expo

nent

s

minus1

minus25

minus2

minus15

minus05



0 1 2 3 4 5 6 7 8 9 10

0

02

b

Lyap

unov

expo

nent

s

minus08

minus06

minus04

minus02


0

05

1

15

2

25

3

35

4

x

b


(a)

0

05

1

15

2

25

3

35

4

45

x

b

2 4 6 81 3 5 7

(b)



01 015 02 025 03

0

05

a

Lyap

unov

expo

nent

s

minus05

minus1

minus15

minus2

(a)

4 45 5 55 6 65 7 75 8 85 9

0

02

04

F

Lyap

unov

expo

nent

s

minus02

minus04

minus06

minus08

minus1

minus12

(b)


0

05

1

15

2

25

3

35

4

45

a

x

minus0501 012 014 016 018 02 022 024 026 028 03

(a)

0

05

1

15

2

25

3

35

F

x

4 45 5 55 6 65 7 75 8 85 9 95

(b)









4 Conclusion




Acknowledgments


References

















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Taking 119866119892(1199090 119887) = 119879 we can get

119891 (120582) = 1205823+ (2 minus 2119909

0+ 119886) 120582

2

+ (2119886 (1 minus 1199090) + 119891 (119909

0 119887) (1 + 2119879

2)) 120582

+ 119891 (1199090 119887) (119886 + 2119879

2(1 minus 119909

0minus 11988721199090))

(33)

Let 119860 = 2 minus 21199090+ 119886 119861 = 2119886(1 minus 119909

0) + 119891(119909

0 119887)(1 + 2119879

2) and

119862 = 119891(1199090 119887)(119886 + 2119879

2(1 minus 119909

0minus 11988721199090))

Note that the 119891(1199090 119887) gt 0 119892(119909

0 119887) gt 0 1198792 gt 0 and

1199090lt 1 +

119886

2

1199092

0+

4 + 2119865 + 21198651198872

3 (1 + 1198872)

1199090+

1 + 2119865

3 (1 + 1198872)

gt 0

1199093

0minus

5119886 + 21198872minus 1198861198872+ 6

2 (1 + 1198872)

1199092

0

+

9119886 + 6 + 119886119865 + 31198862minus 1198861198872119865

2 (1 + 1198872)

1199090

minus

(1 + 119886) (2 + 2119886 + 119886119865)

2 (1 + 1198872)

lt 0

(34)




2119909 + (1198872+ 1)1199092) = 119866


1199090lt 1 +

119886

2

1199092

0+

4 + 2119865 + 21198651198872

3 (1 + 1198872)

1199090+

1 + 2119865

3 (1 + 1198872)

gt 0

1199093

0minus

5119886 + 21198872minus 1198861198872+ 6

2 (1 + 1198872)

1199092

0

+

9119886 + 6 + 119886119865 + 31198862minus 1198861198872119865

2 (1 + 1198872)

1199090

minus

(1 + 119886) (2 + 2119886 + 119886119865)

2 (1 + 1198872)

= 0

12057311199094

0minus 12057321199093

0+ 12057331199092

0+ 12057341199090+ 1205735= 0

(35)


1205731=

41198862(1 minus 119887

4)

1198662

1205732=

41198862

1198662(2 + 2119887

2+ 119886 + 119886119887

2+ 2119865 minus 2119865119887

4)

1205733=

411988621198652(1 minus 119887

4)

1198662

+

81198651198862(2 + 2119887

2+ 119886 + 119886119887

2)

1198662

+ 3 (1198872+ 1)

1205734= 4119886119887


2minus

41198862119865 (2 + 2119887

2+ 119886 + 119886119887

2)

1198662

1205735= 12119886 + 2119886119865 + 2119886119887

2119865 + 4119886

2+ 3

(36)


1199090)+119891(119909

0 119887)(1+2119879

2) and119862 = 119891(119909

0 119887)(119886+2119879

2(1minus1199090minus11988721199090))

Since

1199090lt 1 +

119886

2

1199092

0+

4 + 2119865 + 21198651198872

3 (1 + 1198872)

1199090+

1 + 2119865

3 (1 + 1198872)

gt 0

1199093

0minus

5119886 + 21198872minus 1198861198872+ 6

2 (1 + 1198872)

1199092

0

+

9119886 + 6 + 119886119865 + 31198862minus 1198861198872119865

2 (1 + 1198872)

1199090

minus

(1 + 119886) (2 + 2119886 + 119886119865)

2 (1 + 1198872)

= 0

(37)


0) = 119862(119909

0)119860(119909

0) gt 0 and


11199094

0minus12057321199093

0+12057331199092

0+12057341199090+1205735= 0


0)1198611015840(1199090) where

1205731=

41198862(1 minus 119887

4)

1198662

1205732=

41198862

1198662(2 + 2119887

2+ 119886 + 119886119887

2+ 2119865 minus 2119865119887

4)

1205733=

411988621198652(1 minus 119887

4)

1198662

+

81198651198862(2 + 2119887

2+ 119886 + 119886119887

2)

1198662

+ 3 (1198872+ 1)

1205734= 4119886119887


2

minus

41198862119865 (2 + 2119887

2+ 119886 + 119886119887

2)

1198662

1205735= 12119886 + 2119886119865 + 2119886119887

2119865 + 4119886

2+ 3

(38)



0) =

119862(1199090)119860(119909

0) gt 0 119862(119909

0) gt 0 and 1198621015840(119909

0) = 119860

1015840(1199090)119861(1199090) minus











2(119886 119887) = 17












119894


119894


119894contained


119894is said to

connect1198631119894and1198632

119894 if 119897 cap 1198631

119894= and 119897 cap 1198632

119894=


1198631

119894and 1198632




119895






0 2 4 6

0

2

4

6

8

G

x

C

B

A


minus2

x = 1

(a)

02

46

8

01

23

45

68

ZY

DC

A

BX024

minus6

minus4

minus2

minus2minus1

(b)



119894and1198632

119894 then


119895and is denoted by119891(119863

119894) sub 119863

119895


119894) and (1198631

119895 1198632

119895)


119895holds for



119890119899119905(119891) ge log119898





1and


1

are(0727230483 1444887781) (0666821561 1360099751)

(0843401487 1245386534) (0913104089 1330174564)

(39)


0

2

4

x A

C

B

minus6

minus4

minus2

0 2 4 6G


minus8

x = 1

(a)

0 1 2 3005

115

ZY

X

0 2

005

115

Z

Y

A

B

C

0

2

4

minus6

minus4

minus2

minus2

minus2minus15minus1

minus05

minus15minus1

minus1

minus05

minus3

(b)


005

1

01

23

45

0123

zy

x

minus1minus05

minus4

minus3

minus2

minus1

(a)

005

1

01

23

45

0123

zy

x

minus1minus05

minus4

minus3

minus2

minus1

(b)


0 02 04 06 08 1 12 14

0

02

04

G

Lyap

unov

expo

nent

s

minus12

minus1

minus08

minus06

minus04

minus02



(1029275093 1245386534) (0964219331 1165586035)

(1085037175 1040897756) (1164033457 1130673317)

(40)






1198675(1198631) 997891997888rarr 119863

1 119867

5(1198631) 997891997888rarr 119863

2

1198675(1198632) 997891997888rarr 119863

1 119867

5(1198632) 997891997888rarr 119863

2

(41)




28

26

24

22

2

18

16

14

12


G

x

(a)

26

24

22

2

18

16

12

14

06 07 08 09 1 11 12 13 14 15

G

x

(b)


minus20

24

minus20

24

0

1

2

3

Z

Y

X

minus1

minus4 minus4

(a)

0 200 400 600 800 1000

0102030405060

Pow

er sp

ectr

um

minus40

minus30

minus20

minus10

10 lowast log10(S)

(b)


0 05 1 15 2 25

0

05

1

15

2

25

x

y

minus25

minus2

minus15

minus1

minus05

minus05

D1

D2

(a)

0 02 04 06 08 1 12 14 16

04

06

08

1

12

14

16

18

x

y

07 08 09 1 111

11

12

13

14

15

Zoom in

D1

D2

D2

D1

D11

D11

D12

D12

D21

D21

D22

H5(D1)

H5(D1)

H5(D2)

H5(D2)

H5(D12)

H5(D11)

H5(D22)

H5(D21)

(b)



0 02 04 06 08 1 12 14 16 18 2

0

05

G

Lyap

unov

expo

nent

s

minus3

minus25

minus2

minus15

minus05

minus1


0 02 04 06 08 1 12 14 16 18 2

0

05

G

Lyap

unov

expo

nent

s

minus1

minus25

minus2

minus15

minus05



0 1 2 3 4 5 6 7 8 9 10

0

02

b

Lyap

unov

expo

nent

s

minus08

minus06

minus04

minus02


0

05

1

15

2

25

3

35

4

x

b


(a)

0

05

1

15

2

25

3

35

4

45

x

b

2 4 6 81 3 5 7

(b)



01 015 02 025 03

0

05

a

Lyap

unov

expo

nent

s

minus05

minus1

minus15

minus2

(a)

4 45 5 55 6 65 7 75 8 85 9

0

02

04

F

Lyap

unov

expo

nent

s

minus02

minus04

minus06

minus08

minus1

minus12

(b)


0

05

1

15

2

25

3

35

4

45

a

x

minus0501 012 014 016 018 02 022 024 026 028 03

(a)

0

05

1

15

2

25

3

35

F

x

4 45 5 55 6 65 7 75 8 85 9 95

(b)









4 Conclusion




Acknowledgments


References

















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











0) =

119862(1199090)119860(119909

0) gt 0 119862(119909

0) gt 0 and 1198621015840(119909

0) = 119860

1015840(1199090)119861(1199090) minus











2(119886 119887) = 17












119894


119894


119894contained


119894is said to

connect1198631119894and1198632

119894 if 119897 cap 1198631

119894= and 119897 cap 1198632

119894=


1198631

119894and 1198632




119895






0 2 4 6

0

2

4

6

8

G

x

C

B

A


minus2

x = 1

(a)

02

46

8

01

23

45

68

ZY

DC

A

BX024

minus6

minus4

minus2

minus2minus1

(b)



119894and1198632

119894 then


119895and is denoted by119891(119863

119894) sub 119863

119895


119894) and (1198631

119895 1198632

119895)


119895holds for



119890119899119905(119891) ge log119898





1and


1

are(0727230483 1444887781) (0666821561 1360099751)

(0843401487 1245386534) (0913104089 1330174564)

(39)


0

2

4

x A

C

B

minus6

minus4

minus2

0 2 4 6G


minus8

x = 1

(a)

0 1 2 3005

115

ZY

X

0 2

005

115

Z

Y

A

B

C

0

2

4

minus6

minus4

minus2

minus2

minus2minus15minus1

minus05

minus15minus1

minus1

minus05

minus3

(b)


005

1

01

23

45

0123

zy

x

minus1minus05

minus4

minus3

minus2

minus1

(a)

005

1

01

23

45

0123

zy

x

minus1minus05

minus4

minus3

minus2

minus1

(b)


0 02 04 06 08 1 12 14

0

02

04

G

Lyap

unov

expo

nent

s

minus12

minus1

minus08

minus06

minus04

minus02



(1029275093 1245386534) (0964219331 1165586035)

(1085037175 1040897756) (1164033457 1130673317)

(40)






1198675(1198631) 997891997888rarr 119863

1 119867

5(1198631) 997891997888rarr 119863

2

1198675(1198632) 997891997888rarr 119863

1 119867

5(1198632) 997891997888rarr 119863

2

(41)




28

26

24

22

2

18

16

14

12


G

x

(a)

26

24

22

2

18

16

12

14

06 07 08 09 1 11 12 13 14 15

G

x

(b)


minus20

24

minus20

24

0

1

2

3

Z

Y

X

minus1

minus4 minus4

(a)

0 200 400 600 800 1000

0102030405060

Pow

er sp

ectr

um

minus40

minus30

minus20

minus10

10 lowast log10(S)

(b)


0 05 1 15 2 25

0

05

1

15

2

25

x

y

minus25

minus2

minus15

minus1

minus05

minus05

D1

D2

(a)

0 02 04 06 08 1 12 14 16

04

06

08

1

12

14

16

18

x

y

07 08 09 1 111

11

12

13

14

15

Zoom in

D1

D2

D2

D1

D11

D11

D12

D12

D21

D21

D22

H5(D1)

H5(D1)

H5(D2)

H5(D2)

H5(D12)

H5(D11)

H5(D22)

H5(D21)

(b)



0 02 04 06 08 1 12 14 16 18 2

0

05

G

Lyap

unov

expo

nent

s

minus3

minus25

minus2

minus15

minus05

minus1


0 02 04 06 08 1 12 14 16 18 2

0

05

G

Lyap

unov

expo

nent

s

minus1

minus25

minus2

minus15

minus05



0 1 2 3 4 5 6 7 8 9 10

0

02

b

Lyap

unov

expo

nent

s

minus08

minus06

minus04

minus02


0

05

1

15

2

25

3

35

4

x

b


(a)

0

05

1

15

2

25

3

35

4

45

x

b

2 4 6 81 3 5 7

(b)



01 015 02 025 03

0

05

a

Lyap

unov

expo

nent

s

minus05

minus1

minus15

minus2

(a)

4 45 5 55 6 65 7 75 8 85 9

0

02

04

F

Lyap

unov

expo

nent

s

minus02

minus04

minus06

minus08

minus1

minus12

(b)


0

05

1

15

2

25

3

35

4

45

a

x

minus0501 012 014 016 018 02 022 024 026 028 03

(a)

0

05

1

15

2

25

3

35

F

x

4 45 5 55 6 65 7 75 8 85 9 95

(b)









4 Conclusion




Acknowledgments


References

















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












0 2 4 6

0

2

4

6

8

G

x

C

B

A


minus2

x = 1

(a)

02

46

8

01

23

45

68

ZY

DC

A

BX024

minus6

minus4

minus2

minus2minus1

(b)



119894and1198632

119894 then


119895and is denoted by119891(119863

119894) sub 119863

119895


119894) and (1198631

119895 1198632

119895)


119895holds for



119890119899119905(119891) ge log119898





1and


1

are(0727230483 1444887781) (0666821561 1360099751)

(0843401487 1245386534) (0913104089 1330174564)

(39)


0

2

4

x A

C

B

minus6

minus4

minus2

0 2 4 6G


minus8

x = 1

(a)

0 1 2 3005

115

ZY

X

0 2

005

115

Z

Y

A

B

C

0

2

4

minus6

minus4

minus2

minus2

minus2minus15minus1

minus05

minus15minus1

minus1

minus05

minus3

(b)


005

1

01

23

45

0123

zy

x

minus1minus05

minus4

minus3

minus2

minus1

(a)

005

1

01

23

45

0123

zy

x

minus1minus05

minus4

minus3

minus2

minus1

(b)


0 02 04 06 08 1 12 14

0

02

04

G

Lyap

unov

expo

nent

s

minus12

minus1

minus08

minus06

minus04

minus02



(1029275093 1245386534) (0964219331 1165586035)

(1085037175 1040897756) (1164033457 1130673317)

(40)






1198675(1198631) 997891997888rarr 119863

1 119867

5(1198631) 997891997888rarr 119863

2

1198675(1198632) 997891997888rarr 119863

1 119867

5(1198632) 997891997888rarr 119863

2

(41)




28

26

24

22

2

18

16

14

12


G

x

(a)

26

24

22

2

18

16

12

14

06 07 08 09 1 11 12 13 14 15

G

x

(b)


minus20

24

minus20

24

0

1

2

3

Z

Y

X

minus1

minus4 minus4

(a)

0 200 400 600 800 1000

0102030405060

Pow

er sp

ectr

um

minus40

minus30

minus20

minus10

10 lowast log10(S)

(b)


0 05 1 15 2 25

0

05

1

15

2

25

x

y

minus25

minus2

minus15

minus1

minus05

minus05

D1

D2

(a)

0 02 04 06 08 1 12 14 16

04

06

08

1

12

14

16

18

x

y

07 08 09 1 111

11

12

13

14

15

Zoom in

D1

D2

D2

D1

D11

D11

D12

D12

D21

D21

D22

H5(D1)

H5(D1)

H5(D2)

H5(D2)

H5(D12)

H5(D11)

H5(D22)

H5(D21)

(b)



0 02 04 06 08 1 12 14 16 18 2

0

05

G

Lyap

unov

expo

nent

s

minus3

minus25

minus2

minus15

minus05

minus1


0 02 04 06 08 1 12 14 16 18 2

0

05

G

Lyap

unov

expo

nent

s

minus1

minus25

minus2

minus15

minus05



0 1 2 3 4 5 6 7 8 9 10

0

02

b

Lyap

unov

expo

nent

s

minus08

minus06

minus04

minus02


0

05

1

15

2

25

3

35

4

x

b


(a)

0

05

1

15

2

25

3

35

4

45

x

b

2 4 6 81 3 5 7

(b)



01 015 02 025 03

0

05

a

Lyap

unov

expo

nent

s

minus05

minus1

minus15

minus2

(a)

4 45 5 55 6 65 7 75 8 85 9

0

02

04

F

Lyap

unov

expo

nent

s

minus02

minus04

minus06

minus08

minus1

minus12

(b)


0

05

1

15

2

25

3

35

4

45

a

x

minus0501 012 014 016 018 02 022 024 026 028 03

(a)

0

05

1

15

2

25

3

35

F

x

4 45 5 55 6 65 7 75 8 85 9 95

(b)









4 Conclusion




Acknowledgments


References

















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

2

4

x A

C

B

minus6

minus4

minus2

0 2 4 6G


minus8

x = 1

(a)

0 1 2 3005

115

ZY

X

0 2

005

115

Z

Y

A

B

C

0

2

4

minus6

minus4

minus2

minus2

minus2minus15minus1

minus05

minus15minus1

minus1

minus05

minus3

(b)


005

1

01

23

45

0123

zy

x

minus1minus05

minus4

minus3

minus2

minus1

(a)

005

1

01

23

45

0123

zy

x

minus1minus05

minus4

minus3

minus2

minus1

(b)


0 02 04 06 08 1 12 14

0

02

04

G

Lyap

unov

expo

nent

s

minus12

minus1

minus08

minus06

minus04

minus02



(1029275093 1245386534) (0964219331 1165586035)

(1085037175 1040897756) (1164033457 1130673317)

(40)






1198675(1198631) 997891997888rarr 119863

1 119867

5(1198631) 997891997888rarr 119863

2

1198675(1198632) 997891997888rarr 119863

1 119867

5(1198632) 997891997888rarr 119863

2

(41)




28

26

24

22

2

18

16

14

12


G

x

(a)

26

24

22

2

18

16

12

14

06 07 08 09 1 11 12 13 14 15

G

x

(b)


minus20

24

minus20

24

0

1

2

3

Z

Y

X

minus1

minus4 minus4

(a)

0 200 400 600 800 1000

0102030405060

Pow

er sp

ectr

um

minus40

minus30

minus20

minus10

10 lowast log10(S)

(b)


0 05 1 15 2 25

0

05

1

15

2

25

x

y

minus25

minus2

minus15

minus1

minus05

minus05

D1

D2

(a)

0 02 04 06 08 1 12 14 16

04

06

08

1

12

14

16

18

x

y

07 08 09 1 111

11

12

13

14

15

Zoom in

D1

D2

D2

D1

D11

D11

D12

D12

D21

D21

D22

H5(D1)

H5(D1)

H5(D2)

H5(D2)

H5(D12)

H5(D11)

H5(D22)

H5(D21)

(b)



0 02 04 06 08 1 12 14 16 18 2

0

05

G

Lyap

unov

expo

nent

s

minus3

minus25

minus2

minus15

minus05

minus1


0 02 04 06 08 1 12 14 16 18 2

0

05

G

Lyap

unov

expo

nent

s

minus1

minus25

minus2

minus15

minus05



0 1 2 3 4 5 6 7 8 9 10

0

02

b

Lyap

unov

expo

nent

s

minus08

minus06

minus04

minus02


0

05

1

15

2

25

3

35

4

x

b


(a)

0

05

1

15

2

25

3

35

4

45

x

b

2 4 6 81 3 5 7

(b)



01 015 02 025 03

0

05

a

Lyap

unov

expo

nent

s

minus05

minus1

minus15

minus2

(a)

4 45 5 55 6 65 7 75 8 85 9

0

02

04

F

Lyap

unov

expo

nent

s

minus02

minus04

minus06

minus08

minus1

minus12

(b)


0

05

1

15

2

25

3

35

4

45

a

x

minus0501 012 014 016 018 02 022 024 026 028 03

(a)

0

05

1

15

2

25

3

35

F

x

4 45 5 55 6 65 7 75 8 85 9 95

(b)









4 Conclusion




Acknowledgments


References

















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










28

26

24

22

2

18

16

14

12


G

x

(a)

26

24

22

2

18

16

12

14

06 07 08 09 1 11 12 13 14 15

G

x

(b)


minus20

24

minus20

24

0

1

2

3

Z

Y

X

minus1

minus4 minus4

(a)

0 200 400 600 800 1000

0102030405060

Pow

er sp

ectr

um

minus40

minus30

minus20

minus10

10 lowast log10(S)

(b)


0 05 1 15 2 25

0

05

1

15

2

25

x

y

minus25

minus2

minus15

minus1

minus05

minus05

D1

D2

(a)

0 02 04 06 08 1 12 14 16

04

06

08

1

12

14

16

18

x

y

07 08 09 1 111

11

12

13

14

15

Zoom in

D1

D2

D2

D1

D11

D11

D12

D12

D21

D21

D22

H5(D1)

H5(D1)

H5(D2)

H5(D2)

H5(D12)

H5(D11)

H5(D22)

H5(D21)

(b)



0 02 04 06 08 1 12 14 16 18 2

0

05

G

Lyap

unov

expo

nent

s

minus3

minus25

minus2

minus15

minus05

minus1


0 02 04 06 08 1 12 14 16 18 2

0

05

G

Lyap

unov

expo

nent

s

minus1

minus25

minus2

minus15

minus05



0 1 2 3 4 5 6 7 8 9 10

0

02

b

Lyap

unov

expo

nent

s

minus08

minus06

minus04

minus02


0

05

1

15

2

25

3

35

4

x

b


(a)

0

05

1

15

2

25

3

35

4

45

x

b

2 4 6 81 3 5 7

(b)



01 015 02 025 03

0

05

a

Lyap

unov

expo

nent

s

minus05

minus1

minus15

minus2

(a)

4 45 5 55 6 65 7 75 8 85 9

0

02

04

F

Lyap

unov

expo

nent

s

minus02

minus04

minus06

minus08

minus1

minus12

(b)


0

05

1

15

2

25

3

35

4

45

a

x

minus0501 012 014 016 018 02 022 024 026 028 03

(a)

0

05

1

15

2

25

3

35

F

x

4 45 5 55 6 65 7 75 8 85 9 95

(b)









4 Conclusion




Acknowledgments


References

















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 02 04 06 08 1 12 14 16 18 2

0

05

G

Lyap

unov

expo

nent

s

minus3

minus25

minus2

minus15

minus05

minus1


0 02 04 06 08 1 12 14 16 18 2

0

05

G

Lyap

unov

expo

nent

s

minus1

minus25

minus2

minus15

minus05



0 1 2 3 4 5 6 7 8 9 10

0

02

b

Lyap

unov

expo

nent

s

minus08

minus06

minus04

minus02


0

05

1

15

2

25

3

35

4

x

b


(a)

0

05

1

15

2

25

3

35

4

45

x

b

2 4 6 81 3 5 7

(b)



01 015 02 025 03

0

05

a

Lyap

unov

expo

nent

s

minus05

minus1

minus15

minus2

(a)

4 45 5 55 6 65 7 75 8 85 9

0

02

04

F

Lyap

unov

expo

nent

s

minus02

minus04

minus06

minus08

minus1

minus12

(b)


0

05

1

15

2

25

3

35

4

45

a

x

minus0501 012 014 016 018 02 022 024 026 028 03

(a)

0

05

1

15

2

25

3

35

F

x

4 45 5 55 6 65 7 75 8 85 9 95

(b)









4 Conclusion




Acknowledgments


References

















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










01 015 02 025 03

0

05

a

Lyap

unov

expo

nent

s

minus05

minus1

minus15

minus2

(a)

4 45 5 55 6 65 7 75 8 85 9

0

02

04

F

Lyap

unov

expo

nent

s

minus02

minus04

minus06

minus08

minus1

minus12

(b)


0

05

1

15

2

25

3

35

4

45

a

x

minus0501 012 014 016 018 02 022 024 026 028 03

(a)

0

05

1

15

2

25

3

35

F

x

4 45 5 55 6 65 7 75 8 85 9 95

(b)









4 Conclusion




Acknowledgments


References

















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













4 Conclusion




Acknowledgments


References

















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Research Article Dynamical Analysis of the Lorenz-84 ...downloads.hindawi.com/journals/jam/2014/296279.pdf · persistent abnormalities of the weather. erefore, it is very important